cfd on unstructured meshes - sorbonne-universite.fr
TRANSCRIPT
CFD on Unstructured Meshes
Olivier Pironneau
November 12, 2010
2
Contents
1 Partial di↵erential equations for fluids 71.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 The Navier-Stokes equations . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Conservation of mass . . . . . . . . . . . . . . . . . . . . . 81.2.2 Conservation of momentum . . . . . . . . . . . . . . . . . 81.2.3 Conservation of energy and and the law of state . . . . . . 9
1.3 Inviscid flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Incompressible flows . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4.1 Boussinesq Approximation . . . . . . . . . . . . . . . . . . 111.5 Potential flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.6 Turbulence modelling . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6.1 The Reynolds Number . . . . . . . . . . . . . . . . . . . . 131.6.2 Reynolds equations . . . . . . . . . . . . . . . . . . . . . . 141.6.3 The k � " model . . . . . . . . . . . . . . . . . . . . . . . 14
1.7 Equations for Compressible Flows . . . . . . . . . . . . . . . . . . 161.7.1 Boundary and initial conditions . . . . . . . . . . . . . . . 18
1.8 Wall-laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.8.1 Generalized wall functions for u . . . . . . . . . . . . . . . 201.8.2 Wall function for the temperature - energy equation . . . . 221.8.3 k and " . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2 Potential Flow 252.1 Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Incompressible Potential Flow . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Generalities. . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.2 Variational formulation and discretisation of (6)-(7) . . . 262.2.3 Complement: Resolution of the linear system by the con-
jugate gradient method. . . . . . . . . . . . . . . . . . . . 302.2.4 Computation of nozzles . . . . . . . . . . . . . . . . . . . . 332.2.5 Computation of the lift of a wing profile . . . . . . . . . . 33
2.3 Potential Subsonic Flows: . . . . . . . . . . . . . . . . . . . . . . 35
3
4
2.3.1 Variational formulation . . . . . . . . . . . . . . . . . . . . 352.3.2 Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . 372.3.3 Fixed Point Algorithm . . . . . . . . . . . . . . . . . . . . 382.3.4 Complement:Resolution by conjugate gradients : . . . . . . 38
3 Convection-Di↵usion Equations 413.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2 Generalities : . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Some theoretical results on the convection-di↵usion equation. 453.3 Time discretization for Convection-Di↵usion . . . . . . . . . . . . 473.4 Time-space Approximation . . . . . . . . . . . . . . . . . . . . . . 523.5 Spacial Approximation . . . . . . . . . . . . . . . . . . . . . . . . 59
3.5.1 The Lax-Wendro↵/ Taylor-Galerkin scheme . . . . . . . . 593.5.2 The streamline upwinding method (SUPG). . . . . . . . . 613.5.3 Upwinding by discontinuity on cells: . . . . . . . . . . . . 62
4 The Stokes Problem 654.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 Functional Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 664.3 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 General Discretization Method . . . . . . . . . . . . . . . . . . . . 684.5 Solution of the Linear Systems . . . . . . . . . . . . . . . . . . . . 70
4.5.1 Resolution of a saddle point problem by the conjugategradient method. . . . . . . . . . . . . . . . . . . . . . . . 71
4.6 Resolution of the saddle point problem by penalization . . . . . . 744.7 Error Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Incompressible Navier-Stokes Equations 775.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Existence, Uniqueness, Regularity. . . . . . . . . . . . . . . . . . . 78
5.2.1 The variational problem . . . . . . . . . . . . . . . . . . . 785.2.2 Existence of a solution. . . . . . . . . . . . . . . . . . . . 795.2.3 Behavior after a long time. . . . . . . . . . . . . . . . . . . 795.2.4 Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Spatial Discretisation . . . . . . . . . . . . . . . . . . . . . . . . 815.3.1 Generalities. . . . . . . . . . . . . . . . . . . . . . . . . . 815.3.2 Error Estimate . . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 Time Discretisation . . . . . . . . . . . . . . . . . . . . . . . . . . 835.4.1 Semi-explicit discretisation in t . . . . . . . . . . . . . . . 835.4.2 Semi-Implicit and Implicit discretisations . . . . . . . . . . 845.4.3 Solution of the non-linear system (59) . . . . . . . . . . . . 85
5
5.5 Discretisation Of The Total Derivative. . . . . . . . . . . . . . . . 875.5.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.6 Other Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.6.1 SUPG and AIE (Adaptive Implicit/Explicit scheme) . . . 90
5.7 Turbulent flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.7.1 Reynolds’ Stress Tensor. . . . . . . . . . . . . . . . . . . . 915.7.2 The Smagorinsky hypothesis . . . . . . . . . . . . . . . . . 935.7.3 The k � ✏ hypothesis . . . . . . . . . . . . . . . . . . . . . 935.7.4 Wall laws . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.7.5 Boundary conditions for the k � ✏ model . . . . . . . . . . 965.7.6 Positivity of ✏ and k. . . . . . . . . . . . . . . . . . . . . . 975.7.7 Numerical Methods. . . . . . . . . . . . . . . . . . . . . . 98
6 Finite Volumes for Euler, Navier-Stokes and Saint Venant’sequations 1016.1 Compressible Euler Equations . . . . . . . . . . . . . . . . . . . . 101
6.1.1 Position of the problem . . . . . . . . . . . . . . . . . . . . 1016.1.2 Bidimensional test problems . . . . . . . . . . . . . . . . . 102
6.2 Finite Volumes and Upwinding by discontinuity . . . . . . . . . . 1036.3 Compressible Navier-Stokes Equations . . . . . . . . . . . . . . . 105
6.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4 Saint-Venant’s Shallow Water Equations . . . . . . . . . . . . . . 106
6.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.2 Numerical scheme in height-velocity formulation . . . . . . 1096.4.3 A numerical scheme in height-flux formulation . . . . . . . 1106.4.4 Comparison of the two schemes . . . . . . . . . . . . . . . 112
6.5 Discontinuous - Galerkin Methods . . . . . . . . . . . . . . . . . . 1126.5.1 SUPG mixed with Lesaint’s method [258] . . . . . . . . . . 1136.5.2 Di↵usion Problems . . . . . . . . . . . . . . . . . . . . . . 114
6
These notes come in support of a course taught at the University of Paris VIto Master students in computational mechanics.
The lectures are followed by tutorials using the public domain freefem++.Alternatively one could use openfoam or gerris.
Chapter 1
Partial di↵erential equations forfluids
1.1 Introduction
This chapter is devoted to a short description of the main state equations for fluidflows solved for the direct problem. As the design approach described throughoutthe book is universal, these equations can be therefore replaced by other partialdi↵erential equations. However, as we will see, the knowledge of the physic ofthe problem and the accuracy of the solution are essential in shape design usingcontrol theory. In particular, we develop wall function modelling considered asreduced order models in incomplete sensitivity evaluation through the book.
1.2 The Navier-Stokes equations
Denote by ⌦ the region of space (R3) occupied by the fluid. Denote by (0, T 1)the time interval of interest. A Newtonian fluid is characterized by
• a density field ⇢(x, t);
• a velocity vector field u(x, t);
• a pressure field p(x, t);
• a temperature field T (x, t);
for all (x, t) 2 ⌦⇥ (0, T 1).
7
8
1.2.1 Conservation of mass
The variation of mass of fluid in O has to be equal to the mass flux across theboundaries of O. So if n denotes the exterior normal to the boundary @O of O,
@tZ
O ⇢ = �Z
@O ⇢u · n,
By using the Stokes formulaZ
Or.(⇢u) =Z
@O ⇢u · n
and the fact that O is arbitrary, the continuity equation is obtained:
@t⇢+r.(⇢u) = 0. (1.1)
1.2.2 Conservation of momentum
The forces on O are the external forces f (gravity for instance) and the forcethat the fluid outside O exercises, �n � pn per volume element, by definition ofthe stress tensor �. Hence Newton’s law, written for a volume element O of fluidgives
Z
O ⇢du
dt=
Z
@O(�n� pn).
Now
du
dt(x, t) = lim
�t!0
1
�t[u(x+ u(x, t)�t, t+ �t)� u(x, t)]
= @tu+X
j
uj@ju ⌘ @tu+ uru
) ⇢(@tu+ uru) +rp�r.� = f.
By the continuity equation, this equation is equivalent to themomentum equation:
@t(⇢u) +r.(⇢u⌦ u) +r.(pI� �) = f. (1.2)
Newtonian flow
To proceed further an hypothesis is needed to relate the stress tensor � to u. ForNewtonian flows � is assumed linear with respect to the deformation tensor:
� = µ(ru+ruT ) + (⇠ � 2µ
3)Ir.u (1.3)
9
The scalars µ and ⇠ are called the first and second viscosities of the fluid. Forair and water the second viscosity ⇣ is very small, so ⇠ = 0 and the equation ofmomentum becomes
@t(⇢u) +r.(⇢u⌦ u) +rp�r.[µ(ru+ruT )� 2µ
3Ir.u] = f. (1.4)
1.2.3 Conservation of energy and and the law of state
Conservation of energy is obtained by writing that the variation of the totalenergy in a volume element balances heat variation and the work of forces O.
The energy E(x, t) per unit mass in a volume element O is the sum of theinternal energy e and the kinetic energy u2/2.
The work done by the forces is the integral over O of u · (f + � � pI)n.By definition of the temperature T , if there is no heat source (combustion...)
the amount of heat received (lost) is proportional to the flux of the temperaturegradient, i.e. the integral on @O of rT.n. The scalar is called the thermalconductivity. So the following equation is obtained
d
dt
Z
O(t)⇢E =
Z
O{@t⇢E +r.[u⇢E]} =
Z
O u · f +Z
@O[u(� � pI)� rT ]n. (1.5)
With the continuity equation and the Stokes formula it is transformed into
@t[⇢E] +r.(u[⇢E + p]) = r.(u� + rT ) + f · u. (1.6)
To close the system a definition for e is needed. For an ideal fluid, such as air andwater in non-extreme situations, Cv and Cp being physical constants, we have
e = CvT, E = CvT +u2
2, (1.7)
and the equation of state
p
⇢= RT, (1.8)
where R is an ideal gas constant. With � = Cp/Cv = R/Cv + 1, the above canbe written as
e =p
⇢(� � 1). (1.9)
10
With the definition of �, the equation for E becomes what is usually referred asthe energy equation:
@t[⇢u2
2+
p
� � 1] +r.{u[⇢u
2
2+
�
� � 1p]} (1.10)
= r.{rT + [µ(ru+ruT )� 2
3µIr.u]u}+ f · u. (1.11)
By introducing the entropy :
s ⌘ R
� � 1log
p
⇢�, (1.12)
another form of the energy equation is the entropy equation:
⇢T (@ts+ urs) =µ
2|ru+ruT |2 � 2
3µ|r.u|2 + �T.
1.3 Inviscid flows
In many instances viscosity has a limited e↵ect. If it is neglected, together withthe temperature di↵usion, ( = 0, ⌘ = ⇠ = 0) the equations for the fluid becomethe Euler equations:
@t⇢+r.(⇢u) = 0 (1.13)
@t[⇢u2
2+
p
� � 1] +r.{u[⇢u
2
2+
�
� � 1p]} = f · u. (1.14)
Notice also that, in the absence of shocks , the equation for the entropy (1.12)becomes
@s
@t+ urs = 0,
hence s is constant on the lines tangent at each point to u (stream-lines). In facta stream-line is a solution of the equation :
x0(⌧) = u(x(⌧), ⌧)
and so
d
dts(x(t), t) =
@s
@xi
@xi
@t+@s
@t= @ts+ urs = 0.
If s is constant and equal to s0 constant at time 0 and if s is also equal to s0 onthe part of � where u · n < 0, then there is an analytical solution s = s0.
11
Finally there remains a system of two equations with two unknowns, forisentropic flows
@t⇢+r.(⇢u) = 0, (1.15)
⇢(@tu+ uru) +rp = f, (1.16)
where p = C⇢� and C = es0 ��1
R .
1.4 Incompressible flows
When the variations of ⇢ are small (water for example or air at low velocity) wecan neglect its derivatives. Then the general equations become the incompressibleNavier-Stokes equations
@tu+ uru+rp� ⌫�u = f/⇢, r.u = 0, (1.17)
with ⌫ = µ/⇢ the kinematic viscosity and p ! p/⇢ the reduced pressure. Ifbuoyancy e↵ects are present in f , we need an equation for the temperature.
An equation for the temperature T and an analytic expression for ⇢ functionof T can be obtained from the energy equation
@tT + urT �
⇢Cv�T =
⌫
2Cv|ru+ruT |2. (1.18)
1.4.1 Boussinesq Approximation
When the temperature varies a little (|T � T0| << T0) then ⇢ varies a little too.The Boussinesq approximation assumes a linear law: ⇢(T ) = ⇢(T0)(1+↵(T�T0)).Then 1
⇢(T ) ⇡ 1⇢(T0)
(1 � ↵(T � T0)). Under gravity f = �ge3 =: ~g, the equationsbecome:
@tu+ uru+rp� ⌫�u =~g
⇢0(1� ↵(T � T0)), r.u = 0,
@tT + urT �
⇢0Cv�T = 0. (1.19)
1.5 Potential flows
For suitable boundary conditions the solution of the Navier-Stokes equations canbe irrotational
r⇥ u = 0.
12
By the theorem of De Rham there exists then a potential function such that
u = r'.
Using the identities :
�u = �r⇥r⇥ u+r(r.u), (1.20)
uru = �u⇥ (r⇥ u) +r(u2
2), (1.21)
we see,that when u is a solution of the incompressible Navier-Stokes equations(1.17) and f = 0, ' is a solution of the Laplace equation :
�' = 0, (1.22)
and the Bernoulli equation, derived from (1.17), gives the pressure
p = k � 1
2|r'|2. (1.23)
This type of flow is the simplest of all.In the same way, with isentropic inviscid flow (1.16)
@t⇢+r'r⇢+ ⇢0�' = 0,
r(',t +1
2|r'|2 + �C⇢0��1⇢) = 0.
If we neglect the convection term r'r⇢ this system simplifies to a non-linearwave equation :
@tt'� c�'+1
2@t|r'|2 = d(t) (1.24)
where c = �C⇢0� is related to the velocity of the sound in the fluid.
Finally, we show that there are stationary potential solutions of Euler equations(1.14) with f = 0. Using (1.20) the equations can be rewritten as
�⇢u⇥r⇥ u+ ⇢ru2
2+rp = 0.
Taking the scalar product with u, we obtain
u · [⇢ru2
2+rp] = 0.
13
Also, the pressure being given by (1.16)
u · (⇢ru2
2+ ⇢��1C�r⇢) = 0.
Or equivalently
u⇢.r(u2
2+ C
�
� � 1⇢��1) = 0.
So the quantity between the parenthesis is constant along the stream lines; thatis we have
⇢ = ⇢0(k � u2
2)
1��1 .
Indeed the solution of the PDE ur⇠ = 0, in the absence of shocks, is ⇠ constanton the stream-lines. If it is constant upstream (on the inflow part of the boundary, u ·n < 0), and if there are no closed stream-lines then ⇠ is constant everywhere.Thus if ⇢0 and k are constant upstream then r⇥ u is parallel to u and, at leastin 2 dimensions , this implies that u derives from a potential. The transonicpotential flow equation. is then obtained
r.[(k � |r'|2) 1��1r'] = 0. (1.25)
The time-dependent version of this equation is obtained from
u = r', ⇢ = ⇢0(k � (u2
2+ @t')
1��1 , @t⇢+r · (⇢u) = 0.
1.6 Turbulence modelling
In this section we consider the Navier-Stokes equations for incompressible flows(1.17).
1.6.1 The Reynolds Number
let us rewrite (1.17) in non-dimensional form.Let U be a characteristic velocity of the flow under study (for example, one
of the non-homogeneous boundary conditions). Let L be a characteristic length(for example, the diameter of ⌦) and T1 a characteristic time (which is a prioriequal to L/U). Let us put
u0 =u
U; x0 =
x
L; t0 =
t
T1.
14
Then (1.17) can be rewritten as
rx0 · u0 = 0,L
T1U@t0u
0 + u0rx0u0 +1
U2rx0p� ⌫
LU�x0u0 = f
L
U2.
So, if we put T1 = L/U, p0 = p/U2, ⌫ 0 = ⌫/LU, then the equations are the samebut with ”prime” variables. The inverse of ⌫ 0 is called the Reynolds number.
Re =UL
⌫.
1.6.2 Reynolds equations
Consider (1.17) with random initial data, u0 = u0 + u00, where u stands for theexpected value .
Taking the expected value of the Navier-Stokes equations leads to
r · u = 0, (1.26)
@tu+r · (u+ u0)⌦ (u+ u0) +rp� ⌫�u = f, (1.27)
which is alsor · u = 0, R = �u0 ⌦ u0, (1.28)
@tu+r · (u⌦ u) +rp� ⌫�u = f +r ·R. (1.29)
1.6.3 The k � " model
Reynolds hypothesis is that the turbulence in the flow is a local function ofru+ruT :
R(x, t) = R(ru(x, t) +ruT (x, t)). (1.30)
If the turbulence is locally isotropic at scales smaller than those described bythe model (1.29) and if the Reynolds hypothesis holds , then it is reasonable toexpress R on the basis formed with the powers of ru + ruT and to relate thecoe�cients to the two turbulent quantities used by Kolmogorov to characterizehomogeneous turbulence: the kinetic energy of the small scales k and their rateof viscous energy dissipation "
k =1
2|u0|2, " =
⌫
2|ru0 +ru0T |2. (1.31)
For two dimensional mean flows (for some ↵(x, t) )
R = ⌫T (ru+ruT ) + ↵I, ⌫T = cµk2
"(1.32)
15
and k and " are modeled by
@tk + urk � cµ2
k2
"|ru+ruT |2 �r.(cµ
k2
"rk) + " = 0, (1.33)
@t"+ ur"� c12k|ru+ruT |2 �r.(c"
k2
"r") + c2
"2
k= 0, (1.34)
with cµ = 0.09, c1 = 0.126, c2 = 1.92, c" = 0.07.
The model is derived heuristically from the Navier-Stokes equations with thefollowing hypotheses:
• Frame invariance and 2D mean flow, ⌫t a polynomial function of k, ".
• u02 and |r⇥ u0|2 are passive scalars when convected by u+ u0 .
• Ergodicity allows statistical averages to be replaced by space averages .
• Local isotropy of the turbulence at the level of small scales.
• A Reynolds hypothesis for r⇥ u0 ⌦r⇥ u0.
• A closure hypothesis : |r⇥r⇥ u0|2 = c2 "2/k.
The constants cµ, c", c1, c2 are chosen so that the model reproduces
• The decay in time of homogeneous turbulence
• The measurements in shear layers in local equilibrium
• The log-wall law in boundary layers .
The model is not valid near solid walls because the turbulence is not isotropicso the near wall boundary layers are removed from the computational domain.An adjustable artificial boundary is placed parallel to the walls � at a distance�(x, t) 2 [10, 100]⌫/u⌧ .A possible set of boundary conditions is then
u, k, " given initially everywhere,u, k, " given on the inflow boundaries at all t,⌫T@nu, ⌫T@nk, ⌫T@n" given on outflow boundaries at all t
u · n = 0,u · s
q⌫|@nu|
� 1
�log(�
s1
⌫|@nu|) + � = 0 on �+ �,
k|�+� = |⌫@n(u · s)|c�12
µ , "|�+� =1
��|⌫@n(u · s)| 32 ,
16
where � = 0.41, � = 5.5 for smooth walls, n, s are the normal and tangent to thewall and � is a function such that at each point of �+ �,
10q⌫/|@n(u · s)| � 100
q⌫/|@n(u · s)|.
These wall functions are classical and are only valid for regions where the flowis attached and the turbulence fully developed in the boundary layer. Below, weshow how to derive generalized wall functions valid up to the wall and also validfor separated and unsteady flows.
1.7 Equations for Compressible Flows
For compressible flows, let us consider the conservation form of the Navier-Stokes equations with the k � " model. As it has been, the model is derivedby splitting the variables into mean and fluctuating parts and use Reynoldsaverages for density and pressure and Favre averages for other variables. Thenon-dimensionalized Reynolds averaged equations are closed by an appropriatemodelling [2], we have:
@t⇢+r · (⇢u) = 0, (1.35)
@t(⇢u) +r · (⇢u⌦ u) +r(p+2
3⇢k) = r · ((µ+ µt)S), (1.36)
@t(⇢E) +r · ((⇢E + p+5
3⇢k)u) = r · ((µ+ µt)Su) +r((�+ �t)rT ), (1.37)
with� =
�µ
Pr, �t =
�µt
Prt, (1.38)
� = 1.4, P r = 0.72 and Prt = 0.9 , (1.39)
where µ and µt are the inverse of the laminar and turbulent Reynolds numbers.In what follows, we call them viscosity. The laminar viscosity µ is given bySutherland’s law :
µ = µ1(T
T1)1.5(
T1 + 110.4
T + 110.4), (1.40)
where f1 denotes a reference quantity for f or its value at infinity if the flow isuniform there and
S = ru+ruT � 2
3r · u I
is the deformation tensor.We consider the state equation for a perfect gas:
p = (� � 1)⇢T (1.41)
17
Experience shows that almost everywhere ⇢k << p, we therefore drop theturbulent energy contributions in terms with first order derivative (the hyperbolicpart) . This improves also the numerical stability, reducing the coupling betweenthe equations [2].
The k�" model [3] we use is classical; it is an extension to compressible flowsof the incompressible version [4]:
@t⇢k +r.(⇢uk)�r((µ+ µt)rk) = Sk, (1.42)
and@t⇢"+r.(⇢u")�r((µ+ c"µt)r") = S". (1.43)
The right-hand sides of (1.42)-(1.43) contain the production and the destructionterms for ⇢k and ⇢":
Sk = µtP � 2
3⇢kr.u� ⇢", (1.44)
vue la def de S ca marche pas il y a un µT en trop
S" = c1⇢kP � 2c13cµ
⇢"r.u� c2⇢"2
k. (1.45)
The eddy viscosity is given by :
µt = cµ⇢k2
". (1.46)
The constant cµ, c1, c2, c" are respectively 0.09, 0.1296, 11/6, 1/1.4245 andP = S : ru.
The constant c2 and c" are di↵erent from their original values of 1.92 and1/1.3.
The constant c2 is adjusted to reproduce the decay of k in isotropic turbulence.With u = 0, ⇢ = ⇢1, T = T1 the model gives
k = k0(1 + (c2 � 1)"0k0
t)�1
c2�1 . (1.47)
The experimental results of Comte-Bellot [5] give a decay of k in t�1.2 and thisfixes c2 = 11/6 while c2 = 1.92 leads to a decay in t�1.087 and therefore to anoverestimation of k.
This has also been reported in [6], where the author managed to computethe right recirculating bubble length for the backward step problem using thestandard k � " model with this new value c2 = 11/6, wall-laws and c" = 1/1.3.
Finally, the compatibility relation between the k � " constants which comesfrom the requirement of a logarithmic velocity profile in the boundary layer [2]
18
gives the c" constant:
c" =1
2pcµ(c2cµ � c1) =
1
1.423, = 0.41, (1.48)
to be compared to the classical value of c" = 1/1.3.
1.7.1 Boundary and initial conditions
The previous system of Navier-Stokes and k � " equations is well posed in thesmall, as the mathematicians say, meaning that the solution exists for a smalltime interval at least, with the following set of boundary conditions:
Inflow and outflowThe idea is to avoid boundary layers such that all second order derivatives
are removed and that the remaining system (Euler-k � " model) is a system ofconservation laws no longer coupled (as we dropped the turbulent contributionsto first order derivative terms). Inflow and outflow boundary conditions areof characteristic types. Where roughly the idea is to impose the value of avariable if the corresponding wave is entering the domain following the sign ofthe corresponding eigenvalue (in 3D):
�1 = u.n+ c,�2,3,4 = u.n,�5 = u.n� c,�6,7 = u.n, (1.49)
where n is the unit outward normal. However, as the system cannot be fullydiagonalized, we use the following approach [10]. Along these boundaries thefluxes are split in positive and negative parts following the sign of the eigenvaluesof the Jacobian A of the convective operator F .
Z
�1F.nd� =
Z
�1(A+Win + A�W1).nd�, (1.50)
where Win is the computed (internal) value at the previous iteration and W1 theexternal value, given by the flow.
SymmetryHere again the idea is to avoid boundary layers. We drop terms with second
order derivatives and the slipping boundary condition (u.n = 0) is imposed inweak form .
Solid wallsThe physical boundary condition is a no-slip boundary condition for the
velocity (u = 0) and for the temperature, either an adiabatic condition (@nT = 0)or an isothermal condition (T = T�). However, the k�" model above is not valid[1] near walls because the turbulence is not isotropic at small scales. In the wall-laws approach the near-wall region is removed from the computational domain
19
and the previous conditions are replaced by by Fourier conditions of the type
u�.n = 0, u�.t = f1(@nu�, @nT�), T� = f2(@nu�, @nT�) (1.51)
for isothermal walls. This will be described in more details later.Initial conditionsFor external as well as internal flows, the initial flow is taken uniform with
small values for k0 and "0 (basically 10�6|u0|). We take the same value for k and" leading to a large turbulent time scale k/" = 1 which characterizes a laminarflow.
u = u0, ⇢ = ⇢0, T = T0, k = k0, " = "0. (1.52)
Internal flow simulations often also require given profiles for some quantities. Thisis prescribed on the corresponding boundaries during the simulation.
1.8 Wall-laws
The general idea in wall-laws is to remove the sti↵ part from boundary layers,replacing the classical no-slip boundary condition by a more sophisticated relationbetween the variables and their derivatives. We introduce a constant quantitycalled friction velocity from:
⇢wu2⌧ = (µ+ µt)
@u
@y|y=� = µ
@u
@y|y=0, (1.53)
where w means at walls and � at a distance � from the real wall. Using u⌧ weintroduce a local Reynolds number:
y+ =⇢wyu⌧
µw. (1.54)
The aim is now to express the behavior of u+ = u/u⌧ in term of y+ which meansthat the analysis will be independent of the Reynolds number .
In this section we describe our approach to wall-laws. We also give anextension to high speed separated flows with adiabatic and isothermal walls. Theingredients are:
• Global wall-laws: numerically valid up to the wall (i.e. 8y+ � 0).
• Weak formulation: pressure e↵ects are taken into account in the boundaryintegrals which come for the integrations by parts.
• Small � in wall-laws: this means that the computational domain should notbe too far from the wall.
20
• Fine meshes: in the sense that the computational mesh should be fineenough for the numerical results to become mesh independent.
• Compressible extension: laws valid for a large range of Mach number.
An important and interesting feature of wall-laws is that they are compatiblewith explicit time scheme, something which is not so real with low-Reynoldscorrections.
1.8.1 Generalized wall functions for u
The first level in the modelling for wall-laws is to consider flows attached (i.e.without separations) on adiabatic walls (i.e . @T
@n = 0). We are looking for lawsvalid up to the wall (i.e. valid 8y+). We consider the following approximatedmomentum equation in near-wall regions (x and y denote the local tangential andnormal directions) :
@y((µ+ µt)@yu) = 0, (1.55)
whereµt =
p⇢⇢wyu⌧ (1� e�y+/70), with y+ =
⇢wu⌧y
µw, (1.56)
is a classical expression for the eddy viscosity valid up to the wall. (1.55) meansthat the shear stress along y is constant. u⌧ is a constant called the frictionvelocity and is defined by :
u⌧ = ((µ+ µt)
⇢w
@u
@y)1/2 = constant, (1.57)
subscript w means at the wall.
High-Reynolds regions
In high Reynolds regions the eddy viscosity µt = p⇢⇢wyu⌧ dominates the
laminar one and this leads to the log-law
@u
@y=
s⇢w⇢
u⌧
y, u = u⌧
s⇢w⇢(1
log(y) + C), (1.58)
provided that @⇢@y << @u
@y which is acceptable because @p@y ⇠ 0 and @T
@y = 0 as the
wall is adiabatic. Therefore @⇢@y ⇠ 0 .
21
We can see that at this level, there is no explicit presence of the Reynoldsnumber. The dependency with respect to the Reynolds number is in the constantC. To have an universal expression, we write,
u = u⌧
s⇢w⇢(1
log(
yu⌧⇢wµw
) + �), (1.59)
where � = � log(u⌧⇢w/µw) +C) is found to have a universal value of about 5 forincompressible flows [8]:
u+ =u
u⌧=
s⇢w⇢(1
log(y+) + 5). (1.60)
Note that we always use wall values to reduce y-dependency as much aspossible. This is important for numerical implementation.
Low-Reynolds regions
In low-Reynolds regions, (1.56) is negligible and (1.55) gives a linear behavior foru vanishing at walls :
⇢wu2⌧ = µ
@u
@y⇠ µ
u
y. (1.61)
In other words, we have:
u+ =u
u⌧=
yu⌧⇢wµw
= y+. (1.62)
General expression
To have a general expression, we define the friction velocity u⌧ as solution of
u = u⌧
s⇢w⇢f(u⌧ ). (1.63)
where f is such that w = u⌧
q⇢w
⇢ f(u⌧ ) is solution of (1.55-1.56). The wall-function
therefore is not known explicitly and depends on density distribution. A hierarchyof laws can be obtained therefore by taking into account compressibility e↵ectsstarting from low-speed laws (see appendix to this chapter). Our aim during thisdevelopment is to provide laws easy to implement for unstructured meshes. Forlow-speed flows, where density variations are supposed negligible, a satisfactorychoice for f is the non-linear Reichardt function fr defined by :
fr(y+) = 2.5 log(1 + y+) + 7.8(1� e�y+/11 � y+
11e�0.33y+), (1.64)
22
This expression fits both the linear and logarithmic velocity profiles and also itsbehavior in the bu↵er region .
1.8.2 Wall function for the temperature - energy equation
Consider the viscous part of the energy equation written in the boundary layer(i.e. @x << @y):
@
@y(u(µ+ µt)
@u
@y) +
@
@y((�+ �t)
@T
@y) = 0.
When we integrate this equation between the fictitious wall (y = �) and the realone (y = 0), we obtain :
(�+ �t)@T
@y|� ��@T
@y|0= u(µ+ µt)
@u
@y|0 �u(µ+ µt)
@u
@y|� . (1.65)
So, thanks to @T@y |0= 0 and u |0= 0 :
(�+ �t)@T
@y|� +u(µ+ µt)
@u
@y|�= 0. (1.66)
Therefore, in the adiabatic case , there is no term for the energy equation toaccount for [11, 12] while for isothermal walls we need to close (1.67) providingan expression for either the first term in the left-hand side or for the right handside (see appendix to this chapter): indexIsothermal walls
(�+ �t)@T
@y|� +u(µ+ µt)
@u
@y|�= �
@T
@y|0 . (1.67)
Remark 2.1 As a consequence, to evaluate the heat transfer at the wall, wehave to use the following formula:
Ch =�@yT |0⇢1u3
1�=
(�+ �t)@yT |� +u⇢wu2⌧
⇢1u31�
.
This is important as industrial codes usually do the post-processing in a separatelevel than computation and the fluxes are not communicated between the twomodules. In other words, with these codes, when using wall functions as well aswith low-Reynolds models, only the first term is present in a heat flux evaluationabove. This might also explain some of the reported weakness of wall functionsfor heat transfer
Remark 2.2 In separation and recirculation areas u and u⌧ needed by ourwall-laws are small. As a consequence, this leads to an underestimation of the
23
heat flux. In these area, by dimension argument, we choose the local velocityscale to be:
u = c�3/4µ
pk. (1.68)
And redefine, the friction flux by:
(µ+ µt)@u
@y= c�3/4
µ (µ+ µt)@pk
@y. (1.69)
1.8.3 k and "
Once u⌧ is computed, k and " are set to:
k =u2⌧pcµ↵ , " =
k32
l", (1.70)
where ↵ = min(1, (y+
20 )2) reproduces the behavior of k when � tends to zero ( �
is the distance of the fictitious computational domain from the solid wall). Thedistance � is given a priori and is kept constant during the computation. l" is alength scale containing the damping e↵ects in the near wall regions.
l" = c�3/4µ y(1� exp(
�y+
2c�3/4µ
)). (1.71)
Again, here the limitation is for separation points where the friction velocity goesto zero while high level for k would be expected.
24
Chapter 2
Potential Flow
2.1 Orientation
We have seen in chapter 1 that it is possible (with suitable boundary conditions)to find irrotational solutions (r⇥u = 0) to the general fluid mechanics equationsunder the following hypotheses :
- inviscid flow- no vorticity generated by discontinuities (shocks) or by the boundaries.In this chapter, we will study finite element approximations of those equations.
The first part deals with the Neumann problem, where we recall also the finiteelement method. In the second part, we deal with subsonic compressible flows,which will be solved by an optimization method. We then extend the methodto transonic flows. Finally in the third and fourth parts we study the resolutionof the problem in terms of stream function and a rotational correction methodbased on Helmoltz decomposition of vector fields.
2.2 Incompressible Potential Flow
2.2.1 Generalities.
If the density ⇢ is constant, the viscosity and thermal di↵usion are negligible andif the flow does not depend on time then the velocity u(x) and pressure p(x) aregiven for all points x 2 ⌦ of the fluid by
r.u = 0 (1)
r⇥ u = 0 (2)
p = k � 1
2u2 (3)
25
26
where k is a constant if the flow is uniform at infinity.On the boundary � = @⌦ in general the normal component of the velocity,
u.n, is given by :
u.n = g on � (4)
We remark that (2) implies the existence of a potential ' such that :
u = r' (5)
So the complete system is simply rewritten as a Neumann problem.
�' = 0 in ⌦ (6)
@'
@n= g on � (7)
Remark 1 We could have used (1) to say that there exists such that
u = r⇥ (9)
Then, the complete system becomes
r⇥r⇥ = 0 in ⌦ (10)
r⇥ .n = g on � (11)
If the flow is bidimensional (invariant by translation in one direction) then (10)-(11) become :
� = 0 (12)
(x(s)) =Z s
g(x(�)d� 8x(s) 2 � (13)
We study this method at the end of the chapter. Note that in 3 dimensions, theoperator in (10)-(11) is not strongly elliptic and that is a vector though ' is ascalar.
2.2.2 Variational formulation and discretisation of (6)-(7)
Proposition 1 Let � 2 L2(⌦)with non zero average on ⌦. If g is regular,(g 2 H1/2(�))and if
Z
�g = 0 (14)
then problem (6)-(7) is equivalent to the following variational problem :
27
Z
⌦r'rw =
Z
�gw 8w 2 W (15)
' 2 W = {w 2 H1(⌦);Z
⌦�w = 0} (16)
Proof : We see, in multiplying (6) by w, and integrating on ⌦ with the help of Green’sformula that :
Z
⌦�(�')w +
Z
�
@'
@nw =
Z
⌦r'rw. (17)
If we use (6) and (7) in (17) we get
Z
⌦r'rw =
Z
�gw 8w 2 H1(⌦) (18)
Putting w = 1 we get (14). As (18) doesn’t change if we replace ' by ' + constant and w byw + constant, we can restrict ourselves to W . The converse can be proved in the same way ;from (15) with the assumption that ' is regular (in H2(⌦)) so as to use (17), we get :
Z
⌦�(�')w +
Z
�(@'
@n� g)w = 0 8w 2 W (19)
By taking w to be zero on the boundary, we deduce that :
Z
⌦�(�')w = 0 8w such that
Z
⌦�w = 0,
and so, from the theory of Lagrange multipliers, there exists a constant � such that
��' = �� in ⌦. (20)
Note also that (19) implies
@'
@n= g on � (21)
Finally, (17) with w = 1 (20), (21) and (14) imply that the constant in (20) is zero.
Remark 2 W given by (16) is equivalent to H1(⌦)/R. One could have setup the problem in H1(⌦)/R at the start but this presentation allows a naturalconstruction for the discretisation of W . Besides, it is interesting to note that if(14) is not satisfied, ' is the solution of (20) instead of (6).
Proposition 2 If ⌦ is a bounded domain with Lipschitz boundary and if g is inH1/2(�) then (15) has a unique solution.
Proof :see [1]Proposition 3Let {Wh} be a sequence of internal approximations (Wh ⇢ W ) of W, such that
28
8w 2 W 9{wh}h, wh 2 Wh such that ||wh � w||1 ! 0 when h ! 0 (24)
Then the solution 'h 2 Wh ofZ
⌦r'hrwh =
Z
�gwh 8wh 2 Wh (25)
converges strongly in H1(⌦) to ', the solution of (15)-(16).Proof : see [1]Proposition 4If there exist ↵ and C(⌦) such that in addition to (24) we have
||wh � w||1 Ch↵||w||↵+1 (29)
then ||'h � '||1 C 0h↵||'||↵+1 (30)
Proof : see [1]
Lagrangian triangular finite elements
(also called P 1 conforming)⌦ is divided into triangles (tetrahedra in 3D) {Tk}1...K such that
• Tk \ Tl = ↵, or 1 vertex, or 1 whole side (resp. side or face) when k 6= l
• The vertices of the boundary of [Tk are on �
• The singular points of � (corners) are on the boundary �h of [Tk.
We note that ⌦h = [Tk, �h = @ [ Tk , {qi}N1 are the vertices of the triangles,and h is the longest side of a triangle :
h = max{i,j,k:qi,qj2T
k
}|qi � qj| (32)
Hh = {wh 2 C0(⌦h) : wh|Th
2 P↵} (33)
Wh = {wh 2 Hh :Z
⌦h
�wh = 0} (34)
where P↵ denotes the space of polynomials in n variables of degree less than orequal to ↵ (⌦ ⇢ Rn) and C0 the space of continuous functions.
If ⌦ is a polygon then Wh ⇢ W and proposition 3 applies. If moreover, weassume that all the angles of the triangles are bounded by ✓1 > 0 and ✓2 < ⇡when h ! 0 then the proposition 4 can be applied.
29
With (33) Wh is of finite dimension, say M-1. Let {w0i}1...M�1 be a basis ofWh ; if we write 'h in this basis, we have
'h(x) =M�1X
1
'iw0i(x) (35)
and if we replace w0h by w0j in (25) we get a (M � 1)⇥ (M � 1) linear system.
A� = G (36)
with � = {'1..'M�1}, Aij =Z
⌦rw0irw0j, Gj =
Z
�gw0j (37)
Example 1With ↵ = 1 a basis of Wh can easily be constructed. Let {�ki (x)}i=1...n+1
denote the barycentric coordinates of x in Tk with respect to its n + 1 verticesthen the number of basis functions is N � 1 (i.e. M = N) that is one less thanthe total number of vertices {q}i=1...N . Let wi be the canonical basis function ofLagrangian elements of order 1:
wi(x) = �ki (x) if k is such that x 2 Tk, (qi 2 Tk) (38)
= 0 otherwise
then
w0i(x) = wi(x)�R⌦ �w
i(x)R⌦ �
i = 1..N � 1 (39)
is a basis of Wh.Remark :As w0i di↵ers from wi only by a constant, thanks to (14), one can use either wi
or w0i in (37). It makes no di↵erence. The e↵ect of this construction is simply topull out a function wi to construct a basis. This procedure has the inconvenienceof distinguishing a vertex from others (since we have pulled the wi associated toit) and sometimes introduces a local numerical error around the pulled vertexbecause the linear system is poorly conditioned there. We will see below that theconjugate gradient method avoids this inconvenience.
Example 2With ↵ = 2 the number of basis functions is equal to the number of vertices
qi plus the midpoints of the sides (edges) minus one. We can also construct itby (39) but with �ik replaced by �ik(2�
ik � 1) for the basis functions associated to
vertices and by 4�ik�jk for the basis functions associated to the midpoints of the
sides (edges).
30
2.2.3 Complement: Resolution of the linear system by theconjugate gradient method.
The linear system (36) obtained by the finite element method has a peculiar structure whichwe must exploit to optimize the resources of the computer. The conjugate gradient method(cf. Polak [193], Lascaux-Theodor [142], Luenberger [162] for example) makes use of thesparse structure of the linear system to speed the solution when properly preconditioned. Thisalgorithm applies also to positive semi-definite linear systems; thus we can construct the system(36), (37) with all the basis functions wi. The solution obtained is more regular
Let us recall the conjugate gradient method briefly.Notation:
Let x 2 RN be an unknown such that Ax = b , where A 2 RN⇥N , AT = A, b 2 RN .Let C be a positive definite matrix and < , >
C
the scalar product associated with C:
< a, b >C
= aTCb; ||a||C
= (aTCa)12 (40)
Algorithm 1
0. Initialization:Choose C 2 RN⇥N positive definite (preconditioning matrix ), ✏ small positive, x0such
that Cx0 2 ImA (0 for example), and set g0 = h0 = �C�1(Ax0 � b), n = 0.1 Calculate :
⇢n =< gn, hn >
C
hnTAhn
(41)
xn+1 = xn + ⇢nhn (42)
gn+1 = gn � ⇢nC�1Ahn (43)
�n =||gn+1||2
C
||gn||2C
(44)
hn+1 = gn+1 + �nhn (45)
2 If || gn+1||C
< ✏ , stop else increment n by 1 and go to 1.Note that C is never inverted and that y = C�1
z is a short hand notation for ”solve Cy =
z”
Remark 1
This algorithm can be viewed as a particular case of a more general algorithm used to findthe minimum of a function (see below). Here it is applied to the computation of the minimumof E(x) = xTAx/2� bTx . One can verify that
i) ⇢n given by (1) is also the minimum of E(xn + ⇢hn)ii) �gn+1 given by (43) is also the gradient of E with respect to the scalar product associated
with C, that is :
gn+1 = �C�1(Axn+1 � b) (46)
Proof :
xn+1 = xn + ⇢nhn <=> Axn+1 = Axn + ⇢nAhn
, C�1(Axn+1 � b) = C�1(Axn � b) + ⇢nC�1Ahn
Remark 2
31
The only divisions in the algorithm are by hnTAhn and ||gn||C
. The first one could be zeroif the kernel of A is non empty whereas the second is non zero by construction; so to prove thatthe algorithm is applicable even if det(A) = 0 we have to prove that hnTAhn is never zero. Butbefore that let us show convergence.
Lemma
hjTAhk = 0, 8j < k (47)
< gk, gj >C
= 0, 8j < k (48)
< gk, hj >C
= 0, 8j < k (49)
Proof :
Let us proceed by the method of induction. Assuming that the property is true forj < k n, let us prove that (47)(48)(49) are true for all j < k n+ 1 .
i) Multiplying (43) by hj and using (47) and (49):
< gn+1, hj >C
=< gn, hj >C
�⇢n < C�1Ahn, hj >C
= 0� hjTAhn = 0, (50)
if j < n ; If j = n then it is zero by (41).ii) Using (45):
< gn+1, gj >C
=< gn+1, hj � �jhj�1 >C
= 0 (51)
by i) above.iii)Finally, again from (45) we have, if j < n :
hn+1TAhj = (gn+1 + �nhn)TAhj = gn+1TAhj + 0 = gn+1TC(gj+1 � gj)
⇢j= 0 (52)
where we have used (47) to get the second equality and (43) for the last one.If j = n, we have to use the definition of �n. Let us show that �n is also equal to
�gn+1TAhn/hnTAhn. We have :
gn+1TAhn
hnTAhn
=< gn+1 � gn, gn+1 >
C
< gn+1 � gn, hn >C
=�||gn+1||2
C
< gn, hn >C
(53)
We have used (43) for the first equation and (48) and (49) for the second.We leave to the reader the task of showing that (47),(49) are true for k = 1. We end the
proof by showing that hnTAhn is never zero.Indeed if n is the first time it is zero then we have:
0 = hnTAhn, hn 2 ImA ) hn = 0 ) gn = ��n�1hn�1
= gn�1 � ⇢n�1C�1Ahn�1
so by (41)��n�1|hn|2 =< gn�1hn�1 > �⇢n�1hn�1TAhn�1 = 0
but by hypothesis hn�1 is not zero.Corollary
The algorithm converges in N iterations at the most.
Proof :
Since the gn are orthogonal there cannot be more than N of them non zero. So at iterationn = N , if not before, the algorithm produces gn = 0 that is C�1(Axn � b) = 0.
32
Choice of C
However it is out of question to do N iterations because N is a very big number. One canprove the following result :
Proposition 5:
If x⇤is the solution and xk
is the computed solution at the kth iteration, then
< xk � x⇤, A(xk � x⇤) >C
4(µC
A
� 1
µC
A
+ 1)2k < x0 � x⇤, A(x0 � x⇤) >
C
where µC
A
is the condition number of A (ratio between the largest and smallest eigenvalues of
Az = �Cz ) in the metric introduced by C.
Proof : see Lascaux-Theodor [9], for example.
One can also prove superconvergence results, that is the sequence {xk} converges fasterthan all the geometric progressions (faster than rk for all r) but this result assumes that thenumber of iterations is large with respect to N .
We can easily verify that if x0 = 0 and C = A we get the solution in one iteration. So thisis an indication that one should choose C ’near’ to A. When we do not have any informationabout A, experience shows that the following choices are good in increasing order of complexityand performance:
Cij
= Aii
�ij
(54)
Cij
= Aij
, 8i, j |i� j| < 2 (55)
Cij
= 0, otherwise.
C = incompletetely factorized matrix of A. (56)
We recall the principle of incomplete factorization (Meijerink-VanderVorst [172], Glowinski etal [97]):
We construct the Choleski factorization L0 of A (= L0L0T ) and put
Lij
= 0 if Aij
= 0, Lij
= L0ij
otherwise.
One can also construct directly L instead of L0by putting to 0 all the elements of C 0 whichcorrespond to a zero element of A, during the factorization (cf [10]) but then the final matrixmay not be positive definite.
Proposition 6
If A is positive semi definite, b is in the image of A, and Cx0is in the image of A, then
the conjugate gradient algorithm converges towards the unique solution x0of the linear system
Ax = b; which verifies Cx0 2 Im A.
Proof :
From (42) and (45) we see that
Cxn, Cgn, Chn 2 ImA ) Cxn+1, Cgn+1, Chn+1 2 ImA
The property is thus proved by induction.
33
2.2.4 Computation of nozzles
If ⌦ is a nozzle we take, in general, � zero on the walls of the nozzle and g = u1.n,u1 constant at the entrance and exit of the nozzle.
The engineer is interested in the pressure on the wall and the velocity field.One sometime solves the same problem with di↵erent boundary conditions at theentrance �1 and at the exit �2 of the nozzle :
��' = 0 in ⌦, '|�1 = 0, '|�2 = constant,@'
@n|@⌦��1[�2 = 0. (57)
The same method applies.
2.2.5 Computation of the lift of a wing profile
The flow around a wing profile S corresponds, in principle, to flow in anunbounded exterior domain, but we approximate infinity numerically by aboundary �1 at a finite distance ; so ⌦ is a two dimensional domain withboundary � = �1 [ S.
One often takes u1 constant and
g|�1 = u1.n, g|S = 0 (58)
Unfortunately, the numerical results show that with these boundary conditions,the flow generally goes around the trailing edge P . As P is a singular point of �,|r'(x)| tends to infinity when x ! P and the viscosity e↵ects (⌘ and ⇣) are nolonger negligible in the neighborhood of P . The modeling of the flow by (1)-(2)is not valid and (2) has to be replaced by (! constant) :
r⇥ u = !�⌃
where �⌃ is the Dirac function on the stream line ⌃ which passes through P .One takes then ⌦ � ⌃ as the domain of computation. So we have to add a
boundary condition on ⌃. Since ⌃ is a stream line :
@'
@n|⌃ = 0. (59)
But as u is continuous along ⌃ we also have :
@'
@�|⌃+ =
@'
@�|⌃� .
by integrating this equation on ⌃ we obtain :
34
@
@�('|⌃+ � '|⌃�) = 0 i.e. for some constant � '|⌃+ � '|⌃� = � (60)
where � is a constant which is to be determined with the help of (59) written onP , or better by :
|r'(P+)|2 = |r'(P�)|2 (61)
which comes out to be the same but is interpreted as a continuity condition onthe pressure.
It can be proved using conformal mappings that with (60)-(61) the solution' does not depend on the position of ⌃ (which is not known a priori) but with(59) this is not the case: the solution depends on the position of ⌃. So we solve
��' = 0 in ⌦� ⌃ (62)
'|⌃+ � '|⌃� = � (63)
|r'(P+)|2 = |r'(P�)|2 (64)
@'
@n|@⌦ = g; (65)
equation (64) which is called the Joukowski condition . It deals with the continuityof the velocity and also of the pressure .
To solve (62)-(65) a simple method is to note that the solution of (62)-(63)-(65) is linear in � :
'(x) = '0(x) + �('1(x)� '0(x)) (66)
- '0 is the solution of (62),(65) and (53) with � = 0, i.e.
�'0 = 0,@'
@n|� = g, ' continuous across ⌃
- = '1 � '0 is the solution of (62),(65) with g = 0 and (63) with � = 1, i.e.
� = 0,@
@n|� = 0, |⌃+ � |⌃� = 1, r |⌃+ = r |⌃�
The variational formulation of this second problem is: find ' 2 W p1 such that
Z
⌦r rw = 0 8w 2 W p
0 ;
W p� = { 2 H1(⌦) : |⌃+ � |⌃� = �, r |⌃+ = r |⌃�}
We find � by solving (64) with (66) : it is an equation in one variable �. Wecan show that the lift Cf (the vertical component of the resultant of the forceapplied by the fluid on S) is proportional to � :
35
Cf = �⇢|u1| (67)
where ⇢ is the density of the fluid.Practical Implementation :In practice we can use P 1 finite elements though the Joukowski condition
requires that we know r' at the trailing edge ; with P 1, r' is piecewise constantand so the triangles should be su�ciently small near the trailing edge. In [11]a rule for refining the triangles near the trailing edge (to keep the error O(h)):can be found the size of the triangles should decrease to zero as a geometricprogression as they approach P and the rate of the geometric progression is afunction of the angle of the trailing edge. Experience shows that one can applythe condition (61) by replacing P+ and P� by the triangles which are on S andhave P as a vertex.
2.3 Potential Subsonic Flows:
2.3.1 Variational formulation
One still assumes that the e↵ects of viscosity and thermal di↵usion are negligiblebut does not assume that the fluid is incompressible. If the flow is stationary,irrotational at the boundary and behind all shocks if any, then one can solve thetransonic potential equation
r.[(k � 1
2|r'|2) 1
��1r'] = 0 in ⌦ (68)
The velocity is still given by
u = r' (69)
the density by
⇢ = ⇢0(k � 1
2|r'|2) 1
��1
and the pressure by
p = p0(⇢
⇢0)� (70)
The constants p0, ⇢0 are usually known at the outer boundary. The boundaryconditions are given on the normal flux ⇢u.n rather than on the normal velocity:
(k � 1
2|r'|2) 1
��1@'
@n= g on � (71)
36
To simplify the notation, let us put
⇢(r') = ⇢0(k � 1
2|r'|2) 1
��1 (72)
Proposition 7Problem (44)-(45) is equivalent to a variational equation. Find
' 2 W = {w 2 H1(⌦);Z
⌦�w = 0}
(where �is any given function with non zero mean on ⌦ ) such that
Z
⌦(k � 1
2|r'|2) 1
��1r'rw =Z
�gw 8w 2 W ; (73)
Moreover, any solution of (49) is a stationary point of the functional
E(') = �Z
⌦(k � 1
2|r'|2) �
��1 � �
� � 1
Z
�g (74)
Proof :
To prove the equivalence between (68) and (73)-(74) we follow the procedure of proposition1. Let us consider now
E('+ �w) =
�Z
⌦(k � 1
2|r('+ �w)|2) �
��1 � �
� � 1
Z
�g('+ �w)
We have :
E0,�
('+ �w) =�
� � 1
Z
⌦(k � 1
2|r('+ �w|2) 1
��1r('+ �w)rw � �
� � 1
Z
�gw
So all the solutions of (49) are such that
E0,�
('+ �w)|�=0 = 0
and now the result follows.
Proposition 8If b < (2k(� � 1)/(� + 1) ) 1/2 then E defined by (70) is convex in
W b = {' 2 W : |r'| b} (75)
Proof :Let us use (71) to calculate E,�� in the direction w :
� � 1
�
d2E
d�2|�=0 =
Z
⌦⇢(r')|rw|2 � 1
(� � 1)
Z
⌦⇢(r')2��(r'.rw)2
37
=Z
⌦⇢(r')2��|rw|2(k � 1
2|r'|2[1 + 2
� � 1(r'.rw
|r'||rw|)2]
�Z
⌦⇢(r')2��|rw|2(k � 1
2|r'|2(1 + 2
� � 1))
� (k � 1
2b2)
2��
��1 (k � b2
2
� + 1
� � 1)||rw||20
Corollary 1If b < (2k (� � 1)/(� + 1) ) 1/2 the problem
min'2W b
�Z
⌦(k � 1
2|r'|2) �
��1 � �
� � 1
Z
�gw (76)
admits a unique solution.Proof :W b is closed convex, E is W � elliptic,convex, continuous.Corollary 2If the solution of (76) is such that |r'| 6= b at all points, then it is also the
solution of (69).
2.3.2 Discretisation
Proposition 9Let b < (2k (� � 1)/(� + 1) ) 1/2 and let
W bh = {'h 2 Wh : |r'h| b} (77)
Assume that W bh and W b satisfy (24) with strong convergence in W 1,1 ; let us
approximate (75) by
min'h
2W b
h
�Z
⌦(k � 1
2|r'h|2)
�
��1 � �
� � 1
Z
�g'. (78)
If {'h}h are solutions in W 1,1then 'h ! ' the solution of (76).Proof :As |r'h| is bounded by b, one can extract a subsequence which converges
weakly in W 1,1(⌦) weak *. Let be its limit. Let ' be the solution of (76) and⇧h' the interpolate of ' in the sense of (24). Let E be the functional of problem(78). Then since W b
h ⇢ W b we have :
E(') E('h) E(⇧h').
The weak semi-continuity of E and the fact that 'h is the solution of (78)implies that any element w of W b is the limit of {wh}, wh 2 W b
h in W 1,1
(limh!0||w � wh||1,1 = 0)
38
E(') E( ) lim inf E('h) limE(⇧h') = E(') (79)
but ' is the minimum and so
E(') = E( ). (80)
The strong convergence of 'h towards ' inH1(⌦) is proved by using the convexityand the W b -ellipticity of E.
2.3.3 Fixed Point Algorithm
The fixed point algorithm, introduced by Gelder [86], when it converges gives theresult very fast :
r.[(k � 1
2|r'n|2) 1
��1r'n+1] = 0 in ⌦ (88)
(k � 1
2|r'n|2) 1
��1@'n+1
@n= g on � (89)
It is conceptually very simple and easy to program. Once discretised by finiteelements (88), (89) gives a linear system at each iteration, but the convergenceis not guaranteed; however experience shows that it works well when the flowis everywhere subsonic. The coe�cients depend on the iteration number n ; soone should avoid direct methods of resolution. If one solves (88) by algorithm 1(conjugate gradient) then it becomes a method which from a practical point ofview is very near to algorithm 2.
2.3.4 Complement:Resolution by conjugate gradients :To treat the constraint ” |r'| < b ” the simplest method is penalization. We then solve :
min'h2Wh
E0('h
) (81)
E0('h
) = �Z
⌦(k � 1
2|r'
h
|2) ���1 � �
� � 1
Z
�g'+ µ
Z
⌦[(b2 � |r'
h
|2)+]2
where µ is the penalization parameter which should be large. The penalization is only to avoidthe divergence of the algorithm if in an intermediate step the bound b is violated. If the solutionreaches the bound b in a small zone then Corollary 2 doesn’t apply any more; (experience showsthat outside of this zone the calculated solution is still reasonable); but in this case it is betterto use augmented Lagrangian methods (cf Glowinski [95]).
Taking into account the fact that Wh
is of finite dimension, (81) is an optimization problemwithout constraint with respect to coe�cients of '
h
on a basis of Wh
:
'h
(x) =N�1X
1
'i
wi(x)
39
min{'1..'N�1}
E0('h
)
To solve (81), we use the conjugate gradient method with a preconditioning constructed froma Laplace operator with a Neumann condition. Let us recall the preconditioned conjugategradient algorithm for the minimization of a functional.
Algorithm 2 (Preconditioned Conjugate Gradients) :
Problem to be solved: minz2R
NE(z) (82)
0. Choose a preconditioning positive definite matrix C; choose ✏ > 0 small, M a large integer.Choose an initial guess z0. Put n = 0.
1. Calculate the gradient of E with respect to the scalar product defined by C; that is thesolution of
Cgn = �rz
E(zn) (83)
If ||gn||C
< ✏ stopelse if n = 0 put h0 = g0 else put
� =||gn||2
C
||gn�1||2C
(84)
hn = gn + �hn�1 (85)
3 . Calculate the minimizer ⇢n solution of
min⇢
{E(zn + ⇢hn)} (86)
Put
zn+1 = zn + ⇢nhn (87)
If n < M increment n by 1 and go to 1 otherwise stop.Proposition 10
If E is strictly convex and twice di↵erentiable, algorithm 2 generates a sequence {zn} which
converges (✏ = 0, M = +1) towards the solution of problem (82).
The proof can be found in Lascaux-Theodor [142] or Polak [193] for example. Let us recallthat convergence is superlinear (the error is squared every N iterations) and that practicalexperiments show that
pN iterations are enough or even more than adequate when we have
found a good preconditioner C. With a good preconditioner, the number of iterations isindependent of N . This is the case with problem (81) when C is the matrix A constructedin (37). The linear systems (83) could be solved by algorithm 1. As in the linear case, thealgorithm works even if the constraint on the mean of '
h
is active provided that the gradientgn is projected in that space.
40
Chapter 3
Convection-Di↵usion Equations
3.1 Introduction
The following PDE for � is called a linear convection-di↵usion equation:
�,t + a�+r.(u�)�r.(⌫r�) = f (1)
where a(x, t) is the dissipation coe�cient (positive), u(x, t) is the convectionvelocity and ⌫(x, t) is the di↵usion coe�cient (positive).
For simplicity we assume that a = 0, as is often the case in practice. All thatfollows, however, applies to the case a > 0; for one thing when a is constant wehave:
(eat�),t = eat(�,t + a�)
so with such a change of function one can come back to the case a = 0; secondlythe di↵usion terms and the dissipative terms have similar e↵ects physically.
These equations arise often in fluid mechanics. Here are some examples :- Temperature equation for ✓ for incompressible flows,- Equations for the concentration of pollutants in fluids,- Equation of conservation of matter for ⇢ and the momentum equation in
the Navier-Stokes equations with � = u although these equations are coupledwith other equations in which one could observe other phenomena than that ofconvection-di↵usion.
In general, ⌫ is small compared to UL ( characteristic velocity ⇥ characteristiclength ) so we must face two di�culties:
- boundary layers,- instability of centered schemes.
On equation (1) in the stationary regime when ⌫ = 0 we shall study first threetypes of methods:
41
42
- characteristic methods,- streamline upwinding methods,- upwinding-by-discontinuity.Then we will study the full equation (1) beginning with the centered schemes
and finally for the full equation, the three above mentioned methods plus theTaylor-Galerkin / Lax-Wendro↵ scheme.
1.1. Boundary layers
To demonstrate the boundary layer problem, let us consider a one dimensionalstationary version of (1) :
u�,x � ⌫�,xx = 1 (2)
�(0) = �(L) = 0 (3)
where u and ⌫ are constants. The analytic solution is
�(x) = (L
u)[x
L� e[(uL/⌫)(x/L)] � 1
e(uL/⌫) � 1] (4)
When ⌫/uL ! 0,
�! (x� L)
uif u < 0, except in x = 0,
�! x
uif u > 0, except in x = L.
But the solution of (2) with ⌫ =0 is
� =(x� L)
u(5)
if we take the second boundary condition in (3) and
� =x
u(6)
if we take the first one.We see that when ⌫/uL ! 0, the solution of (2) tends to the solution with
⌫ = 0 and one boundary condition is lost; at that boundary, for ⌫ << 1, there isa boundary layer and the solution is very steep and so it is di�cult to calculatewith a few discretisation points.
1.2. Instability of centered schemes :
43
Finite element methods correspond to centered finite di↵erence schemes whenthe mesh is uniform and these schemes do not distinguish the direction of flow.
Let us approximate equation (2) by P 1 conforming finite elements. Thevariational formulation of (2)
Z L
0u�,xw + ⌫
Z L
0�,xw,x =
Z L
0w, 8w 2 H1
0 (]0, L[) (7)
is approximated by
Z L
0(u�h,xwh + ⌫�h,xwh,x � wh) = 0 8wh continuous P 1 piecewise (8)
and zero on the boundary.If ]0,L[ is divided into intervals of length h, it is easy to verify that (8) can be
written as
u
2h(�j+1 � �j�1)� ⌫
h2(�j+1 � 2�j + �j�1) = 1, j = 1, ..., N � 1 (9)
where �j = �h(jh), j = 0,...,N = L/h ; �o = �N = 0.Let us find the eigenvalues of the linear system associated with (9), that is,
let us solve
In(�) = det
0
BBBBB@
0
BBBBB@
2↵� � 1� ↵ 0 ...�1� ↵ 2↵� � 1� ↵ ...
... ... ... ...
... �1� ↵ 2↵� � 1� ↵
... 0 �1� ↵ 2↵� �
1
CCCCCA
1
CCCCCA= 0
where we have put ↵ = 2⌫/(hu). One can easily find the recurrence relation forIn :
In = (2↵� �)In�1 + (1� ↵2)In�2
and so when � = 2↵ then I2p+1 = 0 8p. This eigenvalue is proportional to ⌫, andtends to 0 when ⌫ ! 0. So one could foresee that the system is unstable when⌫/hu is very small.
There are several solutions to there di�culties :1. One can try to remove the spurious modes corresponding to null or very
small eigenvalues by putting constraints on the finite element space (see Stenberg[220] for example).
2. One can solve the linear systems by methods which work even for non-definite systems. For example Wornom- Hafez [239] have shown that a block
44
relaxation method with a sweep in the direction of the flow allows (9) to besolved without upwinding.
3. Finally one can modify the equation or the numerical scheme so as toobtain non-singular well posed linear systems: This is the purpose of upwindingand artificial di↵usion.
Many upwinding schemes have been proposed (Lesaint-Raviart [149], Henrichet al. [110], Fortin-Thomasset [83], Baba-Tabata [7], Hughes [115], Benque et al.[20], Pironneau [190] ... see Thomasset [229] for example). We shall study someof them. For clarity we begin with the stationary version of (1).
3.2 Generalities :
In this section, we consider the equation
�,t +r.(u�)� ⌫�� = f in ⌦⇥]0, T [ (53)
�(x, 0) = �o(x) in ⌦; (54)
�|� = �� (55)
and knowing that in general for fluid applications ⌫ is small, we search for schemeswhich work even with ⌫ = 0. Evidently, if ⌫ = 0, (55) should be relaxed to
�|⌃ = �� (56)
If ⌫ = 0, (53), (54), (55) is a particular case of stationary convection (eq (10))for the variables {x, t} on ⌦⇥]0, T [ with velocity {u, 1}.
Thus one could simply add a di↵usion term to the previous schemes and geta satisfying theory. But if we use the ”cylindrical” structure of ⌦⇥]0, T [, we candevise two new methods, one implicit in time without upwinding and the othersemi-implicit but unconditionally stable. We begin by recalling an existence anduniqueness result for (53) and a good test problem to compare the numericalschemes: the rotating hump.
Test Problem.Consider the case where ⌦ is the square ] � 1, 1[2 and where u is a velocity
field of rotation around the origin
u(x, t) = {y,�x}.We take �� = 0, f = 0 and
�0(x) = e(|x�x0|2�r2)�2 8x such that |x� x0| < r,
45
�0(x) = 0 elsewhere.
When ⌫ = 0, �0 is convected by u so if we look at t ! {x, y,�(x, y, t)} we willsee the region {x, y : �(x, y) 6= 0} rotate without deformation; the solution isperiodic in time and we can overlap � and �0 after a time corresponding to oneturn. If ⌫ 6= 0 the phenomenon is similar but the hump flattens due to di↵usion;we may also get boundary layers near the boundary
3.2.1 Some theoretical results on the convection-di↵usionequation.
Proposition 7:Let ⌦ be an open bounded set with boundary � Lipschitz; we denote by n its
exterior normal ; the system
�,t + ur�+ a� = f in Q = ⌦⇥]0, T [ (57)
�(x, 0) = �0(x) 8x 2 ⌦ (58)
�(x, t) = g(x, t) 8(x, t) 2 ⌃ = ((x, t) : u(x, t).n(x) < 0) (59)
has a unique solution in C0(0, T ;L2(⌦)) when �0 2 L2(⌦), g 2 C0(0, T ;L2(�))and a, u 2 L1(Q), Lipschitz in x, f 2 L2(Q).
Proof :We complete the proof by constructing the solution. Let X(⌧) the solution
of
d
d⌧X(⌧) = u(X(⌧), ⌧) if X(⌧) 2 ⌦ (60)
= 0 otherwise
with the boundary condition
X(t) = x. (61)
If u is the velocity of the fluid, then X is the trajectory of the fluid particle thatpasses x at time t. With u 2 L1(Q) problem (60)-(61) has a unique solution. AsX depends on the parameters x, t, we denote the solution X(x, t; ⌧) ; it is alsothe ”characteristic” of the hyperbolic equation (57).
We remark that :
46
d
d⌧�(X(x, t; ⌧), ⌧), ⌧)|⌧=t = �,t(x, t) + ur�(x, t) (62)
So (57) can be rewritten as :
�,⌧ + a� = f (63)
and integrating
�(x, t) = e�R
t
0a(X(⌧),⌧)d⌧ [�+
Z t
0f(X(�), �)e
R�
0a(X(⌧),⌧)d⌧d�] (64)
we determine � from (58) and (59).If X(x, t; 0) 2 �, then
� = g(X(x, t; 0), ⌧(x)) (65)
where ⌧(x) is the time (< t) for X(x, t; ⌧) to reach �.If X(x, t; 0) 2 ⌦, then
� = �o(X(x, t; 0)) (66)
Corollary (a = r.u) :
With the hypothesis of the proposition 7 and if u 2 L2(0, T ;H(div,⌦)) \L1(Q), Lipschitz in x,problem (53), (58), (59) with ⌫ = 0 has a unique solutionin C0(0, T ;L2(⌦)).
-(we recall that H(div,⌦) = {u 2 L2(⌦)n : r.u 2 L2(⌦)}).
Proposition 8 :Let ⌦ be a bounded open set in Rn with � Lipschitz . Then problem
�,t +r.(u�)�r.(r�) = f in Q = ⌦⇥]0, T [ (67)
�(x, 0) = �0(x) in ⌦ (68)
� = g on �⇥]0, T [ (69)
has a unique solution in L2(0, T ;H1(⌦)) if i,j, uj 2 L1(⌦), ui,j 2 L1(⌦),f 2 L2(Q), �o 2 L2(⌦), g 2 L2(0, T ;H1/2(�)), and if there exists a > 0 such that
ijZiZj � a|Z|2 8Z 2 Rn. (70)
Proof :
See Ladyzhenskaya [138].
47
3.3 Time discretization for Convection-Di↵usion
In this paragraph we shall analyze some schemes obtained by using finitedi↵erence methods to discretise @�/@t and the usual variational methods for theremainder of the equation. Let us consider equation (67) with (70) and assume,for simplicity from now on and throughout the chapter that ij = ⌫�ij, ⌫ > 0,u 2 L1(Q) and
r.u = 0 in Q u.n = 0 on �⇥]0, T [ (71)
� = 0 on �⇥]0, T [ (72)
As before, ⌦ is also assumed regular.The reader can extend the results with no di�culty to the case u.n 6= 0, with
the help of paragraph 2.As usual, we divide ]0,T[ into equal intervals of length k and denote by �n(x)
an approximation of �(x, nk).
3.3.1. Implicit Euler scheme :We search �n+1
h 2 Hoh, the space of polynomial functions of degree p on atriangulation of ⌦, continuous and zero on � such that for all wh 2 H0h we have
1
k(�n+1
h � �nh, wh) + (un+1r�n+1
h , wh) + ⌫(r�n+1h ,rwh) = (fn+1, wh) (73)
This problem has a unique solution because this is an N ⇥ N linear system, Nbeing the dimension of H0h :
(A+ kB)�n+1 = kF + I� (74)
where Aij = (wi, wj) + ⌫k(rwi,rwj), Bij = (un+1rwi, wj), Fi = (f, wi),I = (wi, wj) and where {wi} is a basis of H0h and �i the coe�cients of �h
on this basis. This system has a unique solution because the kernel of A+ kB isempty :
0 = T (A+ kB) = TA ) = 0. (75)
Proposition 9 :
If � 2 L2(0, T ;Hp+1(⌦)) and �,t 2 L2(0, T ;Hp(⌦)), we have
(|�nh � �(nk, .)|20 + ⌫k|r(�n
h � �(nk, .))|20)12 C(hp + k) (76)
where p is the degree of the polynomial approximation for �nh.
48
Proof :
a) Estimate of the error in time
Let �n+1 be the solution of the problem discretised in time only :
1
k(�n+1 � �n) + un+1r�n+1 � ⌫��n+1 = fn+1;�n+1|� = 0;�o = 0 (77)
the error ✏n(x) = �n(x) - �(nk, x) satisfies ✏n|� = 0, ✏o = 0 and
1
k(✏n+1 � ✏n) + un+1r✏n+1 � ⌫�✏n+1 = k
Z 1
0(1� ✓)
@2�
@t2(x, (n+ ✓)k)d✓ (78)
where the right hand side is the result of a Taylor expansion of �(nk + k, x). Multiplying (78)by ✏n+1 and integrating on ⌦ makes the convection term disappear and we deduce that
||✏n+1||⌫
⌘ (|✏n+1|20 + k⌫|r✏n+1|20)12 k2T |�00
tt
|0,⌦⇥]0,T [ + ||✏n||⌫
. (79)
So we have, for all n:
||✏n||⌫
kpT |�00
,tt
|0,Q (80)
b) Estimation of the error in space
Let ⇠n = �nh
� �n ; by subtracting (77) from (73) we see
1
k(⇠n+1 � ⇠n, w
h
) + (un+1r⇠n+1, wh
) + ⌫(r⇠n+1,rwh
) = 0 (81)
Let n
h
be the interpolation of �n ; then ⇠n = ⇠nh
+ n
h
� �n and ⇠nh
satisfies (we take wh
=⇠n+1h
)
||⇠n+1h
||2⌫
|⇠nh
|o
|⇠n+1h
|o
� ( n+1h
� �n+1 � n
h
+ �n, ⇠n+1h
)� (82)
�k(un+1r( n+1h
� �n+1), ⇠n+1h
)� ⌫k(r( n+1h
� �n+1),r⇠n+1h
)
so
||⇠n+1h
||⌫
||⇠nh
||⌫
+ Ck[hp||�0,t
||p,⌦ + hp(||u||1,Q
+ ⌫)||�||p+1,⌦] (83)
because |.|o
||.||⌫
, n+1h
- n
h
is the interpolation of �n+1 � �n which is an approximation ofk�0
,t
.Finally we obtain (76) by noting that
||�nh
� �(nk)||⌫
||⇠nh
||⌫
+ || n
h
� �n||⌫
+ ||�(nk)� �n||⌫
(84)
Comments :We note that the error estimate (76) is valid even with ⌫ = 0. Thus, we do not
need any upwinding in space. This is because the Euler scheme itself is upwindedin time (it is not symmetric in n and n+ 1). The constant c in (76) depends on||�||p+1,⌦ and that if ⌫ is small (67) has a boundary layer and ||�||p+1,⌦ tends toinfinity (if �� is arbitrary ) as ⌫�p+1/2. This requires the modification of �� or theimposition of: h << ⌫1�1/2p.
A third method consists of replacing ⌫ by ⌫h in such a way that h << ⌫1�1/2p
is always satisfied. This is the artificial viscosity method.
49
3.3.2. Leap frog scheme :The previous scheme requires the solution of a non-symmetric matrix for each
iteration. This is a scheme which does not require that kind of operation butwhich works only when k is O(h).
1
2k(�n+1
h � �n�1h , wh) + (unr�n
h, wh) +⌫
2(r[�n+1
h + �n�1h ],rwh) (85)
= (fn+1, wh) 8wh 2 H0h, �n+1h 2 H0h
To start (85), we could use the previous scheme.
Proposition 10 :The scheme (85) is marginally stable , i.e. there exists a C such that
|�nh|o C|f |0,Q(1� C|u|1k
h)�
12 (86)
for all k such that
k <h
C|u|1 . (87)
Proof in the case where u is independent of t and f = 0.
We use the energy method (Ritchmeyer-Morton [197], Saiac [211] for example).Let
Sn
= |�n+1h
|20 + |�nh
|20 + 2k(ur�nh
,�n+1h
) (88)
then
Sn
� Sn�1 = |�n+1
h
|20 � |�n�1h
|20 + 2k(ur�nh
,�n+1h
+ �n�1h
) (89)
but from (85), with wh
= �n+1h
+ �n�1h
and if f = 0
|�n+1h
|20 � |�n�1h
|20 + 2k(ur�nh
,�n+1h
+ �n�1h
) + k⌫|r(�n+1h
+ �n�1h
)|20 = 0; (90)
so Sn
Sn�1 ... S
o
. On the other hand, using an inverse inequality
(ur�nh
,�n+1h
) � �C
h|u|1|�n+1
h
|0|�nh
|0 � �|u|1 C
2h(|�n+1
h
|20 + |�nh
|20) (91)
we get
Sn
� (1� Ck|u|1h
)[|�n+1h
|20 + |�nh
|20] (92)
Convergence :As for the Euler scheme, one can show using (86) that (85) is 0(hp + k2) for
the L2 norm error .
50
3.3.3. Adams-Bashforth scheme :It is important to make the dissipation terms implicit as we have done for ��
because the leap-frog scheme is only marginally stable (cf. Richtmeyer-Morton[197]). In the same way, if ⌃ 6= ↵ (u.n 6= 0), it is necessary to make implicitthe integral on ��⌃. For this reason, we consider the Adams-Bashforth schemeof order 3 which is explicit when we use mass lumping and which has a betterstability than the leap-frog scheme.
1
k(�n+1
h � �nh, wh) =
23
12b(�n
h, wh)� 16
12b(�n�1
h , wh) +5
12b(�n�2
h , wh)
8wh 2 H0h where
b(�h, wh) = �[(ur�h, wh) + ⌫(r�h,rwh)� (f, wh)]
The stability and convergence of the scheme can be analysed as in §3.3.5.
3.3.4 The ✓� schemes :
In a general way, let A and B be two operators and the equation in time be
u,t + Au+Bu = f ; u(0) = u0
We consider a scheme with three steps
1
k✓(un+✓ � un) + Aun+✓ +Bun = fn+✓
1
(1� 2✓)k(un+1�✓ � un+✓) + Aun+✓ +Bun+1�✓ = fn+1�✓
1
k✓(un+1 � un+1�✓) + Aun+1 +Bun+1�✓ = fn+1
An analysis of this scheme in the finite element context can be found in Glowinski[96]. One can show easily that with A = ↵C, B = (1�↵)C, where C is an matrixN⇥N with strictly positive eigenvalues, then the scheme is unconditionally stableand of order 2 in k if ↵ = 1/2 and ✓ = 1�p
2/2.In the case of the convection-di↵usion equation, we could take A = �↵⌫�
and B = ur � (1 � ↵)⌫� (↵ = 1 being admissible). Chorin [53], Beale-Majda[17] have studied methods of this type where A = �⌫� and steps 1 and 3 arecarried out by a Monte-Carlo method, and where step 2 is integrated by a finiteelement method, a finite di↵erence method or by the method of characteristics.We will use these for the Navier-Stokes equations.
51
3.3.5. Adaptation of the finite di↵erence techniques :
All the methods presented up to now in this section have been largely studiedon regular grids in a finite di↵erence framework. The same techniques can beused in a finite element context to estimate stability and errors, with the followingrestrictions :
- constant coe�cients (u and ⌫ constants),- uniform triangulation- influence of boundary conditions is di�cult to take into accountThe analysis of finite di↵erence schemes is based on the following fundamental
property:Stability + consistency ) convergence.
To give an example, let us study the Crank-Nicolson scheme for (67) in thecase where u is constant :
1
k(�n+1
h � �nh, wh) +
1
2(ur(�n+1
h + �nh), wh) +
⌫
2(r(�n+1
h + �nh),rwh)
= (fn+ 12 , wh) 8wh 2 H0h
Although the L2 stability is simple to prove by taking wh = �n+1h + �n
h, we willconsider the other methods at our disposal.
By choosing a basis {wi} of H0h we could write explicitly the linear systemcorresponding to the case f = 0 and �h|� = 0.
B(�n+1 � �n) +k
2A(�n+1 + �n) = 0
where Bij = (wi, wj) Aij = ⌫(rwi,rwj) + (urwi, wj)
If we know the eigenvalues and eigenvectors of A in the metric B, i.e. the solutions{�i, i} of �B = A , then by decomposing �n on this basis, we get
(B�)n+1i = (B�)ni
(1� k2�i)
(1 + k2�i)
By asking the amplification factor F = (1-k �i/2)/(1+k �i/2) to be smaller than1 , we can deduce an interval of stability of the method.
Evidently, the smallest real part of the eigenvalues is not known but could benumerically determined in the beginning of the calculation (as in Maday et al.[164] for the Stokes problem).
On the other hand, when H0h is constructed with P 1 � continuous functions,the Crank-Nicolson scheme on a uniform triangulation is identical to the followingfinite di↵erence scheme (exercise) :
52
h2
12[6�n+1
i,j + �n+1i+1,j+1 + �n+1
i+1,j + �n+1i,j�1 + �n+1
i�1,j�1 + �n+1i�1,j + �n+1
i,j+1]
+kh
12[(�n+1
i+1,j+1 � �n+1i�1,j�1)(u1 + u2) + (2u1 � u2)(�
n+1i+1,j � �n+1
i�1,j)
+(2u2 � u1)(�n+1i,j+1 � �n+1
i,j�1)] + ⌫k[4�n+1i,j � �n+1
i+1,j � �n+1i,j+1 � �n+1
i�1,j � �n+1i,j�1]
= idem in �n by changing k to �k.
If there exists solutions of the form :
�nlm = ne2i⇡(lx+my)
they have to satisfy
n+1[h2
6[3 + cos((l +m)h) + cos(lh) + cos(mh)]
+ikh
6[sin((l +m)h)(u1 + u2) + sin(lh)(2u1 � u2) + sin(mh)
(2u2 � u1)] + 2⌫k[2� cos(lh)� cos(mh)]]
= n[ same factor but k ! �k]
So we have a formula for the amplification factor.Finally, it is easy to see that the above finite di↵erence scheme is consistent
to order 2. So we have a presumption of convergence of 0(h2+k2) for the methodon a general triangulation .
3.4 Time-space Approximation
In this section we analyze schemes for the convection-di↵usion equation (53)-(55) which still converge when ⌫ = 0 without generating oscillations. Thisclassification is somewhat arbitrary because the previous schemes can be madeto work when ⌫ = 0 or when ' is irregular. But we have put in this sectionschemes which have been generalized to nonlinear equations (Navier-Stokes andEuler equations for example). Let us list the desirable properties for a scheme towork on nonsmooth functions � :
- convergence in L1 norm,- positivity: ' > 0 ) 'h > 0,- convergence to the stationary solution when t ! 1,
53
- localization of the solution if ⌫ = 0 ( that is to say that the solution shouldnot depend upon whatever is downstream of the characteristic when ⌫ = 0).
4.1. Discretisation of the total derivative:
4.1.1 Discretisation in time.We have seen that if X(x, t; ⌧) denotes the solution of
dX
d⌧(⌧) = u(X(⌧), ⌧); X(t) = x (93)
then
�,t + ur� =@
@⌧�(X(x, t; ⌧), ⌧)|⌧=t (94)
Thus, taking into account the fact that X(x, (n+1)k; (n+1)k) = x, we can write:
(�,t + ur�)n+1 ⇠= 1
k[�n+1(x)� �n(Xn(x))] (95)
where Xn(x) is an approximation of X(x, (n+ 1)k;nk).We shall denote by Xn
1 an approximation 0(k2) of Xn(x) and by Xn2 an
approximation 0(k3) (the di↵erences between the indices ofXand the exponents ofk are due to the fact that Xn is an approximation of X obtained by an integrationover a time k ; thus a scheme 0(k↵) gives a precision 0(k↵+1)).
For example
Xn1 (x) = x� un(x)k (Euler scheme for (93)) (96)
Xn2 (x) = x� un+ 1
2 (x� un(x)k
2)k ( Second order Runge-Kutta) (97)
modified near the boundary so as to get Xni (⌦) ⇢ ⌦. To obtain this inclusion one
can use (96) or (97) inside the elements so that one passes from xto Xn(x) by abroken line rather than a straight line (see (18)(19)).
This yields two schemes for (53) :
1
k(�n+1 � �noXn
1 )� ⌫��n+1 = fn+1 (98)
1
k(�n+1 � �noXn
2 )�⌫
2�(�n+1 + �n) = fn+ 1
2 (99)
Lemma 1 :
54
If u is regular and if Xni (⌦) ⇢ ⌦, the schemes (98) and (99) are L2�stable
and converge in 0(k) and 0(k2) respectively.
Proof :Let us show consider (98). We multiply by �n+1 :
|�n+1|2 + ⌫k|r�n+1|2 (|fn+1|k + |�noX1|)|�n+1| (100)
But the map x ! X preserves the volume when un is solenoidal (r.u = 0). Sofrom (96) :
|�noXn1 |20 =
Z
Xn
1 (⌦)�n(y)2det[rXn
1 ]�1dy |�n|20,⌦(1 + ck2) (101)
Hence �n verifies
||�n||⌫ c[|f |0,Q + |�o|0,⌦] (102)
To get an error estimate one proceeds as in the beginning of the proof ofproposition 9 by using (95).
Remark :Xn
i (⌦) ⇢ ⌦ is necessary because u.n = 0. Otherwise one only needsXn
i (⌦) \ @⌦ ⇢ ⌃.
4.1.2 Approximation in space.Now if we use the previous schemes to approximate the total derivative
(scheme (98) of order 1 , scheme (99) of order 2) and if we discretise in space by aconforming polynomial finite element we obtain a family of methods for which noadditional upwinding is necessary and for which the linear systems are symmetricand time independent .
Take for example the case of (98) :
Z
⌦�n+1h wh + k⌫
Z
⌦r�n+1
h rwh = kZ
⌦fn+1wh +
Z
⌦�nh(X
n1 (x))wh(x)
8wh 2 H0h �n+1h 2 H0h (103)
where H0h is the space of continuous polynomial approximation of order 1 on atriangulation of ⌦, and zero on the boundaries.
Proposition 11 :If X1(⌦) ⇢ ⌦, the scheme (103) is L2(⌦) stable even if ⌫ = 0.
Proof :
55
One simply replaces wh by �n+1h in (103) and derives upper bounds :
|�n+1h |20
Z
⌦|�n+1
h |2 + kZ
⌦⌫r�n+1
h r�n+1h (104)
= kZ
⌦fn+1�n+1
h +Z
⌦�nh(X
n1 (x))�
n+1h (x)
(k|fn+1|0 + |�nhoX
n1 (.)|0)|�n+1
h |0 (|�n
h|0(1 +c
2k2) + k|fn+1|0)|�n+1
h |0The last inequality is a consequence of(101).
Finally by induction one obtains
|�nh|0,⌦ (1 +
c
2k2)n(|�o
h|0,⌦ +X
k|fn|0,⌦) (105)
Remark :
By the same technique similar estimates can be found for (105) but the normson �n
h et �oh will be ||.||⌫ (cf. (79)).
Proposition 12 .If H0h is a P 1 conforming approximation of H1
0 (⌦) then the L2(⌦) norm ofthe error between �n
h solution of (103) and �n solution of (98) is 0(h2/k+h).Thusthe scheme is 0(h2/k + k + h) .
Proof :One subtracts (98) from (103) to obtain an equation for the projected error:
✏n+1h = �n+1
h � ⇧h�n+1, (106)
where ⇧h �n+1 is an interpolation in H0h of �n+1. One getsZ
⌦✏n+1h wh + k⌫
Z
⌦r✏n+1
h rwh �Z
⌦✏nhoX
n1wh = (107)
Z
⌦(�n+1 � ⇧h�
n+1)wh + ⌫kZ
⌦r(�n+1 � ⇧h�
n+1)rwh
�Z
⌦(�n � ⇧h�
n)oXn1wh
From (107), with wh = ✏n+1h we obtain
||✏n+1h ||2⌫ (||✏nh||⌫ + ||�n+1 � ⇧h�
n+1||⌫ + |�n � ⇧h�n|0)||✏n+1
h ||⌫therefore
56
||✏n+1h ||⌫ ||✏nh||⌫ + C(h2 + ⌫kh)
Remark :
By comparing with (103), we see that ✏h and �h are solutions of the sameproblem but for ✏h, f is replaced by :
1
k(�n+1 � ⇧h�
n+1)� ⌫�h(�n+1 � ⇧h�
n+1)� 1
k(�n � ⇧h�
n)oXn1 ,
where �h is an approximation of �. We can bound independently the first andthe last terms. By working a little harder (Douglas-Russell [69]) one can shownthat the error is, in fact, 0(h+ k +min(h2/k, h)).
Proposition 13 :With the second order scheme in time (99) and a similar approximation in
space one can build schemes 0(h2 + k2 + min(h3/k, h2)) with respect to the L2
norm:
(�n+1h , wh) +
⌫k
2(r(�n+1
h + �n),rwh) = (�nhoX
n2 , wh) + k(fn+ 1
2 , wh) (108)
8wh 2 H0h;�n+1h 2 H0h
where H0h is a P 2 conforming approximation of H10 (⌦).
Proof :The proof is left as an exercise.
The case ⌫ = 0 :We notice that (103) becomes
Z�n+1h wh =
Z
⌦�nhoX
n2wh + k
Z
⌦fn+1wh 8wh 2 H0h (109)
�n+1h 2 H0h
That is to say
�n+1h = ⇧h(�
nhoX
n1 ) + k⇧hf
n+1 (110)
where ⇧h is a L2 projection operator in W0h. Scheme (108) becomes :
�n+1h = ⇧h(�
nhoX
n2 ) +
k
2⇧h(f
n+1 + fn) (111)
57
If f = 0 the only di↵erence between the schemes are in the integration formulafor the characteristics. Notice also that the numerical di↵usion comes from theL2 projection at each time step. Thus it is better to use a precise integrationscheme for the characteristics and use larger time steps. Experience shows thatk ⇡ 1.5h/u is a good choice.
Notice that when ⌫ and f are zero one solves
�,t + ur� = 0 �(x, 0) = �0(x) (112)
-(since we have assumed u.n = 0, no other boundary condition is needed).Since r.u = 0, we deduce from (112) that (conservativity)
Z
⌦�(t, x) =
Z
⌦�0(x) 8t (113)
On the other hand, from (109), with wh = 1Z
⌦�n+1h (x) =
Z
⌦�nhoX
n1 =
Z
Xn
1 (⌦)�nh(y)det|rXn
1 |�1dy (114)
So if det|rXn1 | = 1 (which requires Xn
1 (⌦) = ⌦), one hasZ
⌦�n+1h (x)dx =
Z
⌦�nh(y)dy =
Z
⌦�0(x)dx (115)
We say then that the scheme is conservative. It is an important property inpractice.
4.1.3. Numerical implementation problems :
Two points need to be discussed further.- How to compute Xn(x),- How to compute In :
In =Z
⌦�hoX
nhwh (116)
Computation of (116) :
As in the stationary case one uses a quadrature formula:
In ⇠=X
!k�h(Xh(⇠k))wh(⇠
k) (117)
For example with P 1 elements one can takea) {⇠k} = the middles of the sides, !k = �k/3 in 2D, �k/4 in 3D, where �k is
the area (volume) of the elements which contain ⇠k.b) The 3 (4 in 3D) point quadrature formula (Zienkiewicz [241], Stroud [224])
or any other more sophisticated formula ; but experiments show that quadrature
58
formula with negative weights !k should not be chosen. Numerical tests with a4 points quadrature formulae can be found in Bercovier et al. [27].
Finally, another method (referred as dual because it seems that the basisfunctions are convected forward) is found by introducing the following change ofvariable:
Z
⌦�hoXhwh =
Z
Xh
(⌦)�h(y)wh(X
�1h (y))det|rX�1
h |dy (118)
Then the quadrature formula are used on the new integral. If r.u is zero one caneven take
I ⇠=X
!k�h(⇠k)wh(X
�1h (⇠k))wk
h (119)
It is easy to check that this method, (109), (119), is conservative while (109),(117) is not. Numerical tests for this method can be found in Benque et al. [20].
The stability and error estimate when a quadrature formula is used is animportant open problem ; it seems that they have a bad e↵ect in the zones whereu is small (Suli [226], Morton et al.[177]).
Computation of Xn(x) :Formula (96) and (97) can be used directly. However it should be pointed out
that in order to apply (117), (same problem with (119)) one needs to know thenumber l of the element such that
Xn(⇠k) 2 Tl (120)
This problem is far from being simple. A good method is to store all thenumbers of the neighboring (by a side) elements of each element and computethe intersections {⇠k, Xn(⇠k)} with all the edges (faces in 3D) between ⇠k andXn(⇠k); but then one can immediately improve the scheme for Xh by updating uwith its local value on each element when the next intersection is searched; thena similar computation for (18)-(19) is made:
Let {ui} be such that u =P
i uiqi,P
ui = 0, where {qi} are the vertices ofthe triangle (tetrahedron):
Find ⇢ such that
�0i = �i + ⇢ui )Y
i
�0i = 0, �0i � 0. (121)
This is done by trial and error; we assume that it is �m which is zero, so:
⇢ = ��mµm
, (122)
and we check that �i � 0, 8i . If it is not so we change m until it works.
59
Most of the work goes into the determination of the ui. Notice that it may bepractically di�cult to find which is the next triangle to cross when Xn(⇠k) is avertex, for example. This requires careful programming.
When |u � uh|1 is 0(hp) the scheme is 0(hp). If uh is piecewise constant onemust check that uh.n is continuous across the sides (faces) of the elements (whenr.u = 0) otherwise (121) may not have solutions other than ⇢ = 0. If uh = r ⇥ h and h is P 1 and continuous then uh.n is continuous.
3.5 Spacial Approximation
3.5.1 The Lax-Wendro↵/ Taylor-Galerkin scheme
Consider again the convection equation
�,t +r.(u�) = f in ⌦⇥]0, T [ (123)
For simplicity assume that u.n|� = 0 and that u and f are independent of t. ATaylor expansion in time of � gives:
�n+1 = �n + k�n,t +
k2
2�n,tt + 0(k3) (124)
If one computes �,t and �,tt from (123), one finds
�n+1 = �n + k[f �r.(u�n)]� k2
2r.(u[f �r.(u�n)]) + 0(k3) (125)
or again
�n+1 = �n + k[f �r.(u�n)] +k2
2[�r.(uf) +r.[ur.(u�n)]] + 0(k3) (126)
This is the scheme of Lax-Wendro↵ [144]. In the finite element world this schemeis known as the Taylor-Galerkin method (Donea [68]). Note that the last term isa numerical di↵usion 0(k) in the direction un because it is the tensor un ⌦ un.
Let us discretise (126) with Hh , the space of P 1 continuous function on atriangulation of ⌦ :
(�n+1h , wh) = ((�n
h, wh) + k(f �r.(u�nh), wh) (127)
60
�k2
2(r.(uf), wh)� k2
2(urwh,r.(u�n
h)) 8wh 2 Hh
The scheme should be O(h2 + k2) in the L2 norm but it has a CFL stabilitycondition (Courant-Friedrischs-Lewy) (see Angrand-Dervieux [2] for a result ofthis type on an O(h)-regular triangulation for scheme (127) with mass lumping)
k < Ch
|u|1 (128)
An implicit version can also be obtain by changing n into n+1 and k into �k in(126)
(�n+1h , wh) + k(r.(u�n+1
h ), wh) +k2
2(r.(u�n+1
h ), urwh) (129)
= k(f, wh) + (�nh, wh) +
k2
2(r.(uf), wh), 8wh 2 Hh
In the particular case when r.u = 0 and u.n|� = 0,we have the following result :
Proposition 14 :If u and f are independent of t and r.u = 0, u.n|� = 0, then scheme (129)
has a unique solution which satisfies
|�nh|0 |�o
h|0 + T |f |0 + T
2k|r.(uf)|0 (stability) (130)
|�nh � �(nk)|0 C(h2 + k2) (convergence) (131)
Proof :
The symmetric part of the linear system yielded by (129) multiplied left andright by h gives | h|2o + k2/2 |ur h|2 which is always positive ; thus the linearsystems being square, they have one and only one solution.
By taking wh = �n+1h in (129), one finds
|�n+1h |20 +
k2
2|ur�n+1
h |2 |�nh|0|�n+1
h |0 + k|f |0|�n+1|0 + k2
2|r.(uf)|0|�n+1
h |0 (132)
thus if one divides by |�n+1h |o and adds all the inequalities, (130) is found.
To obtain (131) one should subtract (126) from (129) :
(✏n+1, wh) +k2
2(ur✏n+1, urwh) + k(ur✏n+1, wh) = (✏n + 0(k3), wh) (133)
61
where ✏ = �h � �. Let ✏h = �h� h where h is the projection of � in Hh withthe norm ||.||k = (|.|20 + k2/2|ur.|20)|1/2. Then ✏h = ✏ + 0(h2) and from (133), weget
||✏n+1h ||k ||✏nh||+ C(kh2 + k3) (134)
The result follows.
Remarks :1. The previous results can be extended without di�culty to the case ⌫ 6= 0
with the scheme:
(�n+1h , wh) + k(r.(u�n+1
h ), wh) +k2
2(r.(u�n+1
h ), urwh) + ⌫(r�n+1h ,rwh) (135)
= k(f, wh) + (�nh, wh) +
k2
2(r.(uf), wh), 8wh 2 Hoh
Higher order schemes :Donea also report that very good higher order schemes can obtained easily by
pushing the Taylor expansion further; for instance a third order scheme would be(f = 0):
�n+1 � [�n � kr.(u�n) +k2
2r.[ur.(u�n)] +
k2
6r.[ur.(u(�n+1 � �n))]] = 0
3.5.2 The streamline upwinding method (SUPG).
The streamline upwinding method studied in §2.4 can be applied to (123) withoutdistinction between t and x but it would then yield very large linear systems.But there are other ways to introduce streamline di↵usion in a time dependentconvection-di↵usion equation.
The simplest (Hughes [116]) is to do it in space only; so consider
(�n+1h , wh + ⌧r.(uwh)) +
k
2(r.(u[�n
h + �n+1h ]), wh + ⌧r.(uwh))
+k⌫
2(r(�n+1
h + �nh),rwh)� k⌫
2
X
l
Z
Tl
(�(�n+1h + �n
h)⌧r.(uwh))
62
= k(fn+ 12 , wh + ⌧r.(uwh)) + (�n
h, wh + ⌧r.(uwh)), 8wh 2 Hoh
where Tl is an element of the triangulation and u is evaluated at time (n+1/2)kif it is time dependant; ⌧ is a parameter which should be of order h but has thedimension of a time.
With first order elements, ��h = 0 and it was noticed by Tezduyar [227]that ⌧ could be chosen so as to get symmetric linear systems when r.u = 0; anelementary computation shows that the right choice is ⌧ = k/2. Thus in that casethe method is quite competitive, even though the matrix of the linear system hasto be rebuilt at each time step when u is time dependant.
The error analysis of Johnson [125] suggests the use of elements discontinuousin time, continuous in space and a mixture upwinding by discontinuity in timeand streamline upwinding in space.
To this end space-time is triangulated with prisms. LetQn = ⌦⇥]nk, (n+1)k[,let W n
oh be the space of functions in{x, t} which are zero on �⇥]nk, (n + 1)k[continuous and piecewise a�ne in x and in t separately on a triangulation byprisms of Qn ;
We search �nh with �n
h � ��h 2 W noh solution of
Z
Qn
[�nh,t +r(u�n
h)][wh + h(wh,t +r(uwh)]dxdt+Z
Qn
⌫r�nh.rwhdxdt (136)
+Z
⌦�nh(x, (n� 1)k + 0)wh(x, (n� 1)k)dx =
Z
Qn
f(wh + h(wh,t +r.(uwh))dxdt
+Z
⌦�n�1h (x, (n� 1)k � 0)wh(x, (n� 1)k)dx, 8wh 2 W n
oh
Note that if N is the number of vertices in the triangulation of Qn, equation (136)
is an N ⇥N linear system, positive definite but non symmetric.One can show (Johnson et al. [127]) the following :
(Z T
0|�n
h � �|20,⌦dt)12 C(h
32 + k
32 )||�||H2(Q). (137)
3.5.3 Upwinding by discontinuity on cells:
The method of §2.5 can be extended to the nonstationary case but there is anapproximation problem for �� when �h is discontinuous in space. One way is to
63
use the upwinding by cells as in §2.6 ; since the scheme will be 0(h) one can usea piecewise constant approximation in time, which is similar to a discretisationof (53) ( see (49) for the notation) by
(�n+1h , wi)� k
|�i|Z
@��i
u.n[Z
�i
�n+1h dx]d� + ⌫k(r�n+1
h rwi) =
= (�nh, w
i) + k(f, wi), 8i, �n+1h � ��h 2 Hoh
where wi is the continuous piecewise a�ne function associated with the ith vertexof the triangulation of ⌦.
This scheme is used in Dervieux et al [64][65] for the Euler equations (seeChapter 6).
64
Chapter 4
The Stokes Problem
4.1 Problem Statement
The generalized Stokes problem is to find u(x) 2 Rn and p(x) 2 R such that
↵u� ⌫�u+rp = f r.u = 0 in ⌦, (1)
u = u� on � = @⌦. (10)
where ↵, ⌫ are given positive constants and f is a function from ⌦ into Rn.It can come from at least two sources :1) an approximation of the fluid mechanics equations such as that seen in
Chapter 1, that is when the Reynolds number is small (microscopic flow, forexample), then ↵ = 0, and in general f = 0;
2) a time discretisation of the Navier-Stokes equations. Then ↵ is the inverseof the time step size and f an approximation of �uru.
In this chapter, we shall deal with some finite element approximations of (1).More details regarding the properties of existence, unicity and regularity canbe found in Ladyzhenskaya [138], Lions [153] and Temam [228] and regardingthe numerical approximations in Girault-Raviart [93], Thomasset [229], Girault-Raviart [93], Glowinski [95], Hughes [116] .
Test Problem :The most classic test problem is the cavity problem :The domain ⌦ is a square ]0, 1[2 , ↵ = 0, ⌫ = 1, f = 0 and the boundary
condition:u = 0 on all the boundary except the upper boundary ]0, 1[⇥{1} where
u = (1., 0).
65
66
4.2 Functional Setting
Let n be the dimension of the physical space (⌦ ⇢ Rn) and
J(⌦) = {u 2 H1(⌦)n : r.u = 0} (2)
Jo(⌦) = {u 2 J(⌦) : u|� = 0} (3)
Let us put
(a, b) =Z
⌦aibi a, b 2 L2(⌦)n (4)
(A,B) =Z
⌦AijBij A,B 2 L2(⌦)n⇥n (5)
and consider the problem
↵(u, v) + ⌫(ru,rv) = (f, v) 8v 2 Jo(⌦) (6a)
u� u0� 2 Jo(⌦) (6b)
where u0� is an extension in J(⌦) of u�.
Lemma 1If u� 2 H1/2 (�)n and if
R� u�.n =0 then there exists an extension u0
� 2 J(⌦)of u�.
Theorem 1If f 2 L2(⌦)n and u0
� 2 J(⌦), problem (6) has a unique solution.
Proof :Jo(⌦) is a non empty closed subspace of H1(⌦)n and the bilinear form
{u, v} ! ↵(u, v) + ⌫(ru,rv)
is H10 (⌦) elliptic. With the hypothesis v ! (f, v) continuous, the theorem is a
direct consequence of the Lax-Milgram theorem.
4.3 Discretization
Let {Jh}h be a sequence in a finite dimensional space, J0h = Jh \H10 (⌦)
n, suchthat
8v 2 Jo(⌦) there exists vh 2 Joh such that ||vh � v||1 ! 0 when h ! 0 (12)
67
We consider the approximated problem :
Find uh such that
↵(uh, vh) + ⌫(ruh,rvh) = (f, vh) 8vh 2 Joh; uh � u0�h
2 Joh. (13)
where u0�h
is an approximation of u0� in Jh.
Theorem 3If Joh is non empty the problem (13) has a unique solution.
ProofLet us consider the problem
minuh
�u0�h
2Joh
1
2{↵(uh, uh) + ⌫(ruhruh)� (f, uh)} (14)
Since Joh is of finite dimension N , (14) is an optimization of a strictly quadraticfunction in N variables ; thus it admits a unique solution. By writing the firstorder optimality conditions for (14), we find (13).
A counter example .Let ⌦ be a quadrilateral and Th be the triangulation of 4 triangles formed
with the diagonals of ⌦. Let
J0h = {vh 2 C0(⌦) : vh|Tk
2 P 1 vh|� h = 0,
(r.vh, q) = 0 8q continuous, piecewise a�ne on Th}There are only two degrees of freedom for vh (its values at the center) but thereare 4 constraints of which 3 are independent, so J0h is reduced to {0, 0}.
To solve numerically (13) (or (14)) we can try to construct a basis of Joh.Let {vi}Ni be a basis of Joh. By writing uh on this basis,
uh(x) =X
1..N
uivi(x) + u0�h
(x), (15)
it is easy to see by putting (15) in (13) with vh = vi that (13) reduces to a linearsystem
AU = G (16)
with
68
Aij = ↵(vi, vj) + ⌫(rvi,rvj), (17)
Gj = (f, vj)� ↵(u0�h
, vj)� ⌫(ru0�h
,rvj). (18)
However there are two problems
1) It is not easy to find an internal approximation of J(⌦) ; that is in generalwe don’t have Joh ⇢ J0(⌦), which leads to considerable complications in the studyof convergence.
2) Even if we succeed in constructing a non empty Joh which satisfies (12)and for which we could show convergence, the construction of the basis {vi} is ingeneral di�cult or unfeasible which makes (15)(16) inapplicable in practice.
4.4 General Discretization Method
Before solving these problems, let us give some examples of discretisation for Johin increasing order of di�culty. These examples are all convergent and feasibleas we shall see later. They are all of the type
Jh = {vh 2 Vh : (r.vh, q)h = 0 8q 2 Qh}J0h = {vh 2 Jh : vh|� = 0}
Thus J0h is determined by the choice of two spaces1o) Vh which approximate H1(⌦),2o) Qh which approximate L2(⌦) or L2(⌦)/R3o) and the choice of a quadrature formula. (., .)h.
As usual, {Tk} denotes a triangulation of ⌦, ⌦h the union of Tk, {qj}j=1..Ns
the vertices of the triangulation, �kj (x) the j� th barycentric coordinate of x withrespect to vertices {qkj}j=1.,n+1 of the element Tk. We denote respectively by
Ns, the number of vertices,Ne, the number of elements,Nb, the number of vertices on �,Na, the number of sides of the triangulation, andNf , the number of faces of the tetrahedra in 3D.
Pm denotes the space of polynomials (of degree m) in n variables. We recallEuler’s geometric identity :
Ne�Na+Ns = 1 in 2D , Ne�Nf +Na�Ns = �1 in 3D
69
3.1. P1 bubble/P1 element (Arnold-Brezzi-Fortin [5])
Let µk(x) be the bubble function associated with an element Tk defined by :
µk(x) =Y
j=1..n+1
�kj (x) on Tk and 0 otherwise
The function µk is zero outside of Tk and on the boundary and positive in theinterior of Tk. Let wi(x) be a function defined by
wi(x) = �ki (x) on all the Tk which contain the vertex i and 0 otherwise
Let us put :
Vh = {Xj
vjwj(x) +X
k
bkµk(x) : 8vj, bk 2 Rn} (19)
that is the set of (continuous) functions having values in Rn and being the sum ofa continuous piecewise a�ne function and a linear combination of bubbles. Weshall remark that dimVh = (Ns+Ne)n.
Let
Qh = {Xj
pjwj(x) : 8pj 2 R} (20)
that is a set of piecewise a�ne and continuous functions (usual P 1 element). Wehave : dim Qh = Ns
Let us put
Jh = {vh 2 Vh : (r.vh, qh) = 0 8qh 2 Qh} (21)
Joh = {vh 2 Jh : vh|�h
= 0}. (22)
It is easy to show that all the constraints (except one) in (21) are independent.In fact if N is the dimension of Vh the constraints which define Jh are of the formBV = 0 where V are the values of vh at the nodes. So if we show that
(vh,rqh) = 0 8vh 2 Vh ) qh = constant
that proves that Ker BT is of dimension 1 and so the ImB is of dimension N �1,i.e.B is of rank N � 1 which is to say that all the constraints are independentexcept one (the pressure is defined up to a constant).
70
Let ei be the ith vector in the cartesian system. Let us take bk = �k,iei, vj = 0and vh in (19). Then
0 = (vh,rqh) =@qh@xi
|Ti
area(Ti)
(n+ 1)) qh = constant
So we have (n = 2 or 3):
dimJh = (n� 1)(Ns+Ne)n�Ns+ 1 = Ns+ nNe+ 1 (23)
We shall prove that this element leads to an error of 0(h) in the H1norm for thevelocity. Whereas, without knowing a basis (local in x) one cannot solve (13)and one has to use duality for the constraints. A basis, local in x, for Jh is notknown (by local in x, we mean that each basis function vi has a support aroundqi and it is zero far from qi).
3.3. P2/P1 element (Hood-Taylor [113])
Vh = {vh 2 C0(⌦h)n : vh|T
k
2 (P 2)n 8k} (27)
Qh = {qh 2 C0(⌦h) : qh|Tk
2 P 1 8k} (28)
and of course (21)-(22).
This element is 0(h2) for the H1 error and has the same dimension (26)but it gives matrices with bandwidth larger than (24)-(25). The proof of theindependence constraints is the same as above. The dimension of Jh is therefore:
dimJh = 2(Ns+Ne� 1) + 2Ns�Ns+ 1 = 3Ns+ 2Ne� 1
4.5 Solution of the Linear Systems
If we know a basis of Joh and of Jh it is enough to write uh in that basis, constructA and G by (17)-(18) and solve (16). This is a simple and e�cient method butrelatively costly in memory because A is a big band matrix.
To optimize the memory we store only the non zero elements of A in a onedimensional array again denoted by A. I and J are integer arrays where theindices of rows and columns of non zero elements are stored.
m = 0Loop on i and jIf aij 6= 0 , do m = m +1, A(m) = aij ,I(m) = i , J(m) = jend of loop .
71
mMax = m.
This method is well suited particularly to the solution of the linear system
AU = G
by the conjugate gradient method (see Chapter 2) because it needs only thefollowing operations :
given U calculate V ⌘ AU ;
now this operation is done easily with array A, I and J :To get V (i) one proceeds as follows :
Initialize V = 0.For m = 1.. mMax do
V (I(m)) = A(m) ⇤ U(J(m)) + V (I(m))
4.5.1 Resolution of a saddle point problem by the conju-gate gradient method.
All the examples given for Joh are of the type
Joh = {vh 2 Voh : (r.vh, qh) = 0 8qh 2 Qh} (36)
where Voh is the space of functions of Vh zero on the boundaries. An equivalentsaddle point problem associated with (14) is
minuh
�u0�h
2Voh
maxpi
{12↵(uh, uh) +
1
2⌫(ruh,ruh)� (f, uh) +
X
1..M
pi(r.uh, qi)} (37)
where {qi}M is a basis of Qh.
Theorem 4When Joh is given by (36) and is non empty with dimVh = N , dimQh = M
and Voh ⇢ H10 (⌦)
n ( ⌦ is polygonal), the discretised Stokes problem (13) isequivalent to (37). In addition, (37) is equivalent to
↵(uh, vh) + ⌫(ruh,rvh) + (rph, vh) = (f, vh) 8vh 2 Voh (38)
(r.uh, qh) = 0 8qh 2 Qh (39)
72
uh � u0�h
2 Voh, ph 2 Qh
These are the necessary and su�cient conditions for uh to be a solution of (13)
because they are the first order optimality conditions for (14).
Proposition 1If {vi}N is a basis of Vohand {qj}Ma basis ofQh, problem (38)-(39) is equivalent
to a linear system
✓A BBT 0
◆✓UP
◆=
✓G0
◆(41)
with U being the coe�cient of uh expressed on the basis {ui} and P being thecoe�cients of phexpressed on the basis {qi}.
Aij = ↵(vi, vj) + ⌫(rvi,rvj), i, j = 1..N (42)
Bij = (rqi, vj) i = 1..M, j = 1..N (43)
Gj = (f, vj)� ↵(u�h
, vj)� ⌫(ru�h
,rvj) j = 1..N (44)
Proposition 2
Problem (41) is equivalent to
(BtA�1B)P = BtA�1G, U = �A�1BP + A�1G (47)
Algorithm 2 (solving (47) by the Conjugate Gradient Method)
0 . Initialisation: choose p0 (=0 if no initial guess known). ChooseC 2 RN⇥Npositive definite, choose ✏ << 1, nMax >> 1; (pn, qn, z, gn are inthe pressure space while un, v are in the velocity space)
solve Au0 = G� Bp0 (49)
put g0 = Btu0 (= BtA�1G) n = 0. (50)
1 . Solve
Av = Bqn (51)
and Cz = Btv (= BtA�1Bqn) (52)
set ⇢ =|gn|2C
< qn, z >C(53)
73
2. Put
pn+1 = pn � ⇢qn (54)
un+1 = un � ⇢v (so that Aun+1 = �Bpn+1 +G) (55)
gn+1 = gn + ⇢z (56)
3. IF (|gn+1| < ✏)or(n > nMax) THEN stop ELSEPut
� =|gn+1|2c|gn|2c
(57)
qn+1 = gn+1 + �qn (58)
and Return to 1 with n:=n+1 .
Choice of the preconditioning.Evidently the steps (51) and (56) are costly. We note that (51) can be
decomposed into n Dirichlet sub-problems for each of the n components of uh.This is a fundamental advantage of the algorithm because all the manipulation isdone only on a matrix not bigger than a scalar Laplacian matrix.
(56) is also a discrete linear system. A good choice for C (Benque and al [20])can be obtained from the Neumann problem on ⌦ for the operator �� , whenwe use for Qh an admissible approximation of H1(⌦) (Cf. 3.3, 3.2 and 3.1):
(r�h,rqh) = (l, qh) 8qh 2 Qh/R ) (59)
Cij = ��hij ⌘ (rqi,rqj) 8i, j (boundary points included) . (60)
This choice is based on the following observation : from (1) we deduce
��p = r.f in ⌦,@p
@n= f.n+ ⌫�u.n
So in the case ⌫ ⌧ 1 there underlies a Neumann problem.This preconditioning can be improved as shown by Cahouet-Chabard [47] by
taking instead of ��h :
C ⌘ (⌫I�1h � ↵��1
h )�1 with Neumann conditions on the boundary
where Ih is the operator associated with the matrix (wi, wj) where wi is thecanonical basis function associated with the vertex qi. In fact, the operatorBTA�1B is a discretisation of r.(↵I � ⌫�)�1r. Now if ↵ = 0, it is the identityand if ⌫ = 0, it is ��h.
74
4.6 Resolution of the saddle point problem bypenalization
The matrix of problem (41) is not positive definite.Rather let us consider
✓A BBt �✏I
◆✓UP
◆=
✓G0
◆(62)
We can eliminate P with the last equations and there remains :
(A+1
✏BBt)U = G (63)
We solve this system by a standard method since the matrix A + ✏�1BBt issymmetric positive definite, (but is still a big matrix). This method is easy toprogram because it can be shown that (63) is equivalent to
↵(uh, vh) + ⌫(ruh,rvh) +1
✏
X
1..M
(r.uh, qi)(r.vh, q
i) = (f, vh) (64)
8vh 2 Voh uh � u0�h 2 Voh
It is useful to replace ✏ by ✏/�i, �i being the area of the support of qi; thiscorresponds to a penalization with the coe�cients �i in the diagonal of I in (62).Other penalizations have been proposed. A clever one (Hughes et al [118]) is topenalize by the equation for u with v = rq :
(r.uh, qh) = 0
is replaced by
(r.uh, qh) + ✏[(rph,rqh)� (�uh,rqh)� (f,rqh)] = 0
but the term (�uh,rqh) can be dropped because r.uh = 0 (see (129) below).
4.7 Error Estimation
Theorem 8If the triangulation is regular (no angle tends to 0 or ⇡ when h tends to 0),
then the P 1�bubble/P 1 element satisfies the inf-sup condition with �0 independentof h. So we have the following error estimate :
||u� uh||1 + |p� ph|o Ch||u||2. (104)
75
The P 2 � P 1 element satisfies
||u� uh||1 + |p� ph|o Ch2||u||3. (105)
76
Chapter 5
Incompressible Navier-StokesEquations
5.1 Introduction
The Navier-Stokes equations :
u,t +uru+rp� ⌫�u = f (1)
r.u = 0 (2)
govern Newtonian incompressible flows ; u and p are the velocity and pressure.These equations are to be integrated over a domain ⌦ occupied by the fluid,during an interval of time ]0,T[. The data are :
- the external forces f ,- the viscosity ⌫ (or the Reynolds number Re),- the initial conditions at t = 0: u0,- the boundary conditions : u�, for example.
These equations are particularly di�cult to integrate for typical application(high Reynolds numbers) (small ⌫ here) because of boundary layers and turbu-lence. Even the mathematical study of (1)-(2) is not complete ; the uniquenessof the solution is still an open problem in 3-dimensions.
There any many applications of the incompressible Navier-Stokes equations ;for example :
- heat transfer problems (reactors,boilers...)- aerodynamics of vehicles (cars, trains, airplanes)- aerodynamics inside motors (nozzles, combustion chamber...)
77
78
- meteorology, marine currents and hydrology.Many tests problems have been devised to evaluate and compare numerical
methods. Let us mention a few:
a ) The cavity problemThe fluid is driven horizontally with velocity u by the upper surface of a
cavity of size a (cf. figure 1). The Reynolds number is defined by : R = ua/⌫.(cf. Thomasset [229]). The domain of computation ⌦ is bidimensional.
b ) The backward step :The parabolic velocity profile (Poiseuille flow) at the entrance and at the exit
section are given, such that the flux at the entrance is equal to the flux at theexit. The domain is bidimensional and the Reynolds number is defined as u1l0/⌫where l’ is the height of the step and u1 the velocity at the entrance at the centreof the parabolic profile (cf. Morgan et al [174]):
l = 3 L = 22 a = 1 b = 1.5.
c ) The cylinder problem.The problem is two dimensional until the Reynolds number rises above
approximately 200 at which point it becomes tridimensional. The domain ⌦is a periodic system of parallel cylinders of infinite length (for 3D computationsone could take a cylinder length equal to 10 times their diameter ). The integral,Q, of u.n at the entrance of the domain is given and is equal to the total flux ofthe flow. The Reynolds number is defined by 3Q/2⌫ (cf. Ronquist-Patera [202]).
In this chapter, we give some finite element methods for the Navier-Stokesequations which are a synthesis of the methods in chapters 3 and 4. We begin byrecalling the known main theoretical results. More mathematical details canbe found in Lions [153], Ladyzhenskaya [138], Temam [228] ; details relatedto numerical questions can be found in Thomasset [229], Girault-Raviart [93],Temam [228] and Glowinski [95].
5.2 Existence, Uniqueness, Regularity.
5.2.1 The variational problem
We consider the equations (1)-(2), in Q = ⌦⇥]0, T [ where ⌦ is a regular openbounded set in Rn, n = 2 or 3, with the following boundary conditions :
u(x, 0) = u0(x) x 2 ⌦ (3)
79
u(x, t) = u�(x) x 2 �, t 2]0, T [ (4)
We embed the variational form, as in the Stokes problem, in the space Jo(⌦) :
J(⌦) = {v 2 H1(⌦)n : r.v = 0} : (5)
Jo(⌦) = {u 2 J(⌦) : u|� = 0}. (6)
The problem is to find u 2 L2(O, T, J(⌦)) \ L1(O, T, L2(⌦)) such that
(u,t , v) + ⌫(ru,rv) + (uru, v) = (f, v), 8v 2 Jo(⌦) (7)
u(0) = u0 (8)
u� u� 2 L2(O, T, Jo(⌦)) (9)
In (7) (, ) denotes the scalar product in L2 in x as in chapter 4 ; the equality isin the L2 sense in t. We take f in L2 (Q) and u0, u� in J(⌦).
We search for u in L1 in order to ensure that the integrals containing uruexist.
5.2.2 Existence of a solution.
Theorem 1The problem (7)-(9) has at least one solution.
5.2.3 Behavior after a long time.
In general we are interested in the solution of (1)-(4) for large time because inpractice the flow does not seem highly dependent upon initial conditions : theflow around a car for instance does not really depend on its acceleration history.
There are many ways ”to forget” these initial conditions for a flow ; here aretwo examples :
1. the flow converges to a steady state independent of t ;2. the flow becomes periodic in time.
Moreover, the possibility is not excluded that a little ”memory” of initialconditions still remains ; thus in 1 the attained stationary state could changeaccording to initial conditions since the stationary Navier-Stokes equations hasmany solutions when the Reynolds number is large (there is no theorem onuniqueness for ⌫ large [28]). We distinguish other limiting states:
3. quasi-periodic flow : the Fourier transform t ! |u(x, t)| (where x is anarbitrary point of the domain) has a discrete spectrum.
80
4. Chaotic flows with strange attractors: t ! |u(x, t)| has a continuous spec-trum and the Poincare sections (for example the points {u1(x, nk), u2(x, nk)}nfor a given x ) have dense regions of points filling a complete zone of space (incase 1, the Poincare sections are reduced to a point when n is large, in cases of 2and 3, the points are on a curve).
In fact, experience shows that the flows pass through the 4 regimes in the order1 to 4 when the Reynolds number Re increases and the change of regime takesplace at the bifurcation points of the mapping ⌫ ! u, where u is the stationarysolution of (1)-(4). As in practice Re is very large (for us ⌫ is small), 4 dominates.The following points are under study :
a) Whether there exists attractors, and if so, can we characterize any of theirproperties ? (Hausdor↵ dimension, inertial manifold,...cf. Ghidaglia [89], Foiaset al.[79] or Berge et al [29] and the bibliography therein).
b) Does u(x, t) behave in a stochastic way and if so, by which law ? Can wededuce some equations for average quantities such as u, |u|2, |r ⇥ u|2... this isthe problem of turbulence modeling and we shall cover it in a little more detailat the end of the chapter . (cf Lesieur [150], for example and the bibliographyfor more details).
Here are the main results relating to point a).Consider (7)-(9)with u� = 0, f independent of t, and ⌦ a subset of R2. This
system has an attractor whose Hausdor↵ dimension is between cRe4/3 and CRe2
where Re =pf diam(⌦)/⌫ (cf. Constantin and al.[59], Ruelle [206][207]). These
results are interesting because they give an upper bound for the number of pointsneeded to calculate such flows (this number is proportional to ⌫�9/4).
In three dimensions, we do not know how to prove that (7)-(9) with the sameboundary conditions has an attractor but we know that if an attractor exists andis (roughly) bounded by M in W 1,1 then its dimension is less than CM3/4⌫�9/4
(cf.[10]).
5.2.4 Euler Equations
As ⌫ is small in practical applications, it is important to study the limit ⌫ ! 0.If ⌫ = 0 the equations (7)-(9) become Euler’s equations. For the problem tobe meaningful we have to change the boundary conditions ; so we consider thefollowing problem :
u,t +uru+rp = f, r.u = 0 (21)
u(0) = u0 (22)
81
u.n|� = g (23)
u(x) = u�(x), 8x 2 ⌃ = {x 2 � : u(x).n(x) < 0}. (24)
Proposition 1If ⌦ is bidimensional and simply connected, if f 2 L2(Q), u0 2 H1(⌦)2,
g = u�.n and (u�)i are in H1/2 (�) and the integral of g on � is zero, then (21) -(24) has a unique solution in L2(O, T ; H1(⌦)2)\ L1(O, T, L2(⌦)2).
5.3 Spatial Discretisation
5.3.1 Generalities.
The idea is simple: we discretise in space by replacing Jo(⌦) by Joh in (7)-(9):Find uh 2 Jh such that
(uh,t , vh) + (uhruh, vh) + ⌫(ruh,rvh) = (f, vh) 8vh 2 Joh (33)
uh(0) = u0h uh � u�
h
2 Joh (34)
If Joh is of dimension N and if we construct a basis for Joh then (33)-(34) becomesa nonlinear di↵erential system of N equations:
AU 0 +B(U,U) + ⌫CU = G. (35)
One could use existing library programs (LINPAK[152] for example) to solve (35)but they are not e�cient, in general, because the special structure of the matricesis not used. This method is known as the ”Method of Lines”
So we shall give two appropriate methods, taking into account the followingtwo remarks :
1o
) If ⌫ >> 1 (33)-(34) tends towards the Stokes problem studied in Chapter4.
2o
) If ⌫ = 0 (33)-(34) is a non-linear convection problem in Joh. In particularin 2-D the convection equation (27) is underlying the system so the techniquesof chapter 3 are relevant.
As we need a method which could adapt to all the values of ⌫ we shallto takethe finite elements in the family studied in the Chapter 4 and the method for timediscretisation adapted to convection studied in Chapter 3. But before that, let ussee a convergence theorem for the approximation (33), (35).
82
5.3.2 Error Estimate .
Let us take the framework of Chapter 4: J(⌦) is approximated by
Jh = {vh 2 Vh : (r.vh, qh) = 0 8qh 2 Qh} (36)
and Jo(⌦) is approximated by Joh = Jh \H10 (⌦)
n. We put :
Voh = {vh 2 Vh : vh|�h
= 0}. (37)
Assume that {Voh, Qh} is such that- there exists ⇧h : V 2
o = H2(⌦)n \H10 (⌦)
n ! Voh such that
(qh,r.(v � ⇧hv)) = 0 8qh 2 Qh ||v � ⇧hv|| Ch||v||2,⌦. (38)
- there exists ⇡h : H1(⌦) ! Qh such that
|q � ⇡hq|0,⌦ Ch||q||1,⌦ (39)
infqh
2Qh
supvh
2Vo
h
(r.vh, qh)
|qh⇡|o||vh||1 � �
||vh||1 C
h|vh|0 8vh 2 V0h (inverse inequality)
Remark :These conditions are satisfied by the P 1+bubble/P 1element and among others,
the element P 1iso P 2/P 1.
To simplify, let us assume that
uo = 0 and u� = 0. (40)
Theorem 5. (Bernardi-Raugel [31])a)n=2. If the solution of (7)-(9) is in L2(0, T ;H1
0 (⌦)2) \ C0(0, T ;Lq(⌦)2),
q > 2, then problem (33)-(34) has only one solution for h small satisfying
||u� uh||L2(o,T ;H10 (⌦)) C(⌫)h||uru||0,Q (41)
b) n=3. If the norm of ⌫�1u in Co(0, T ;L3(⌦)3) is small and if u 2L2(0, T ;H2(⌦)3) \ H1(0, T ;L2(⌦)3) then (34)-(36) has a unique solution withthe same error estimate as in a).
83
5.4 Time Discretisation
5.4.1 Semi-explicit discretisation in t
Let us take the simplest explicit scheme studied in Chapter 3, that is the Eulerscheme.
Scheme 1:Find un+1
h with un+1h � u�h 2 J0h such that
1
k(un+1
h , vh) = g(vh) 8vh 2 J0h, (51)
g(vh) = �⌫(runh,rvh) + (fn, vh) +
1
k(un
h, vh)� (unhrun
h, vh) (52)
Evidently, k has to satisfy a stability condition of the type
k < C(u, h, ⌫) (53)
As for the Stokes problem (51) is equivalent to
1
k(un+1
h , vh) + (rph, vh) = g(vh) 8vh 2 V0h (54)
(r.un+1h , qh) = 0 8qh 2 Qh; (55)
it is still necessary to solve a linear system to get un+1h ; we use a mass-lumping
formula and the P 1isoP 2/P 1element, with quadrature points at the nodes {qi}iof the finer triangulation; �i denotes the area of support of the P 1 continuousbasis functions, vi, associated to node qi; we replace (54) by
�i3k
un+1i + (rph, v
i) = g(vi) +�i3k
uni �
1
k(un
h, vi) 8i (56)
(r.un+1h , qh) = 0 8qh 2 Qh un+1
h (x) =X
i
uivi(x). (57)
We can now substitute uh in (57) and we obtain
X
i
3k
�i(rph, v
i)(rqh, vi) = G(qh) 8i. (58)
that is a linear system of a Laplacian type.The main drawback of these types of methods is the stability (53).There exist schemes which are almost unconditionally stable such as the
rational Runge-Kutta used by Satofuka [212]:
84
To integrate
V,t = F (V ) (59)
we use
Scheme 2
V n+1 = V n + [2g1(g1, g3)� g3(g1, g1)](g3, g3)�1 (60)
where (.,.) represents the scalar product in the space of V (t) and
g1 = kF (V n)
g2 = kF (V n � cg1)
g3 = bg1 + (1� b)g2
This method is of order 2 if 2bc = �1 and of order 1 if it is not. For a linearequation with constant coe�cients, the method is stable when 2bc �1 (for theproof, it is su�cient to calculate the amplification coe�cient when it is appliedto (59) and F (V ) = FV where F is a diagonal matrix). For our problem, wemust add a step of spatial projection on zero divergence functions of J0h. Thatis, we solve :
�i3k
un+1i + (rph, v
i) =�i3k
vn+1i 8vh 2 V0h (61)
(r.un+1h , qh) = 0 8qh 2 Qh (62)
Numerical experiments using this method for the Navier-Stokes equations discre-tised with the P 1isoP 2/P 1 finite elements can be found in Singh [215].
5.4.2 Semi-Implicit and Implicit discretisations
A semi-implicit discretisation of O(k) for (34)-(36) is
Scheme 3:
1
k(un+1
h , vh) + ⌫(run+1h ,rvh) + (un
hrun+1h , vh) = (fn+1, vh) +
1
k(un
h, vh) (63)
A version O(k2) is easy to construct from the Crank-Nicolson scheme. Thisscheme is very popular because it is conceptually simple and almost fully implicit: yet it is not unconditionally stable and each iteration requires the solution ofa non symmetric linear system. Thus on the cavity problem with 10 ⇥ 10 mesh
85
and k = 0.1 the method works well until ⌫ ⇡ 1/500, beyond which oscillationsdevelop in the flow.
Let us consider the following implicit Euler scheme of order O(k)
Scheme 4:
1
k(un+1
h , vh) + ⌫(run+1h ,rvh) + (un+1
h run+1h , vh) = (fn+1, vh) +
1
k(un
h, vh) (64)
8vh 2 J0h. This scheme is unconditionally stable but we must solve a non-linearsystem.
5.4.3 Solution of the non-linear system (59)
.This non-linear system can be solved by Newton’s method, by a least square
method in H�1 or by the GMRES algorithm :
Newton’s Method
The main loop of the algorithm is as follows :For p = 1..pMax do:1. Find �uh with
(�uh, vh)1
k+ ⌫(r�uh,rvh) + (un+1,p
h r�uh + �uhrun+1,ph , vh) = (65)
�{(un+1,ph , vh)
1
k+ ⌫(run+1,p
h ,rvh) + (un+1,ph run+1,p
h , vh)
�(fn+1, vh)� (unh, vh)
1
k} 8vh 2 Joh �uh 2 Joh (66)
2. Put
un+1,p+1h = un+1,p
h + �uh (67)
But here also, experience shows that a condition between k, h and ⌫ is necessaryfor the stability of the scheme because if ⌫ is very small with respect to h, problem(59) has many branch of solutions or because the convergence conditions forNewton’s method (hessien > 0) are not verified. Again for a cavity with a 10⇥10and k = 0.1 we can use this scheme till Re=1000 approximately. If the term uruis upwinded , one can get a method which is unconditionally stable for all meshk ; this is the object of a few methods studied below.
86
The GMRES algorithm.
The GMRES algorithm (Generalized Minimal RESidual method) is a quasi-Newtonian method deviced by Saad [210]. The aim of quasi-Newtonian methodsis to solve non-linear systems of equations :
F (u) = 0, u 2 RN
The iterative procedure used is of the type :
un+1 = un � (Jn)�1F (un)
where Jn is an approximation of F 0(un). To avoid the calculation of F 0 thefollowing approximation may be used:
D�F (u; v)⌘ F (u+ �v)� F (u)
�⇠= F 0(u)v. (69)
As in the conjugate gradient method to find the solution v of Jv = �F oneconsiders the Krilov spaces :
Kn = Sp{r0, Jr0..., Jn�1r0}where r0 = �F �Jv0, v0 is an approximation of the solution to find. In GMRESthe approximated solution vn used in (69) is the solution of
minv2K
n
||r0 � J(v � v0)||.
The algorithm below generates a quasi-Newton sequence which {un} hopefully(it is only certain in the linear case) will converge to the solution of F (u) = 0 inRN :
Algorithm (GMRES) :
0 . Initialisation: Choose the dimension k of the Krylov space; choose u0.Choose a tolerance ✏ and an increment � , choose a preconditioning matrixS 2 RN⇥N . Put n = 0.
1. a. Compute with (69) r1n = �S�1(Fn + Jnun), w1n = r1n/ ||r1n||, where
Fn = F (un) and Jnv = D�F (un; v).b. For j = 2..., k compute rjn and wj
n from
rjn = S�1[D�F (un;wj�1n )�
j�1X
i=1
hni,j�1w
in]
87
wjn =
rjn||rjn|| .
where hni,j = wiT
n S�1 D�F (un;wjn)
2. Find un+1 the solution of
minv=u
n
+P
k
0aj
wn
j
||S�1F (v)||2
⇠= mina1,a2,...a
k
||S�1[F (un) +kX
j=1
ajD�F (un;wjn)]||2
3. If ||F (un+1)|| < ✏ stop else change n into n+1.
The implementation of Y. Saad also includes a back-tracking procedure forthe case where un departs too far away from the solution. The result is a blackbox for the solution of system of equations which needs only a subprogram tocalculate F (u) given u; there is no need to calculate F 0. Experiments have shownthat this implementation is very e�cient and robust.
5.5 Discretisation Of The Total Derivative.
5.5.1 Generalities
Let us apply the techniques developed in chapter 3:
u,t +uru ⇡ 1
k(un+1 � unoXn)
we obtain the following scheme :
Scheme 5:
1
k(un+1
h , vh) + ⌫(run+1h ,rvh) = (fn, vh) +
1
k(un
hoXnh , vh) 8vh 2 J0h (70)
where Xnh (x) is an approximation of the foot of the characteristic at time nk
which passes through x at time (n+ 1)k (Xnh (x) ⇡ x� un
hk).We note that the linear system in (70) is still symmetric and independent of
n. If Xnh is well chosen, this scheme is unconditionally stable, and convergent in
0(k + h).
Theorem 6 :
88
If
Xnh (⌦) ⇢ ⌦, |det(rXn
h )�1| 1 + Ck, (71)
then scheme 5 is unconditionally stable.
Proof :This depends upon the change of variable y = Xn
h (x) in the integralZ
⌦�(Xn
h (x))dx =Z
Xn
h
(⌦)�(y)det(rXn
h )�1dy (72)
From this we obtain
|unhoX
nh |0,⌦ |det(rXn
h )�1|1,⌦|un
h|0,⌦ (73)
In getting a bound from (70) with vh = un+1h and by using (73) and hypothesis
(71), we obtain
||un+1h ||2⌫ ⌘ |un+1
h |20 + k⌫|run+1h |20 (k|fn|0 + (1 + Ck)|un
h|0)|un+1h |0 (74)
Since |uh|0 ||uh||⌫ , we obtain
||un+1h ||⌫ k|fn|0 + (1 + Ck)||un
h||⌫ ; (75)
or, after summation,
||un+1h ||⌫ k
X
in
|f i|0(1 + Ck)n�i + ||u0h||⌫(1 + Ck)n (76)
and hence the result follows with n T/k.
Remark :If un
h = r⇥ h, with h P 1 piecewise and continuous and if Xnh is the exact
solution of
d
d⌧Xh(⌧) = un
h(Xh(⌧)) (77)
then (71) is satisfied with C = 0. It is theoretically possible to integrate (77)exactly since Xh is then a piecewise straight line but, in practice, the exactcalculation of the integral (un
ho Xnh , vh) is unnecessarily costly ; we use a direct
Gauss formula (Pironneau [190])
(unhoXh, vh) ⇠=
X
i
uh(Xnh (⇠
i))vh(⇠i)⇡i (78)
89
or a dual formula (Benque et al. [22])
(unhoXh, vh) ⇠=
X
i
uh(⇠i)vh(X
n�1h (⇠i))⇡i (79)
Theorem 7 :Assume ⌦ convex, bounded, polygonal and u 2 W 1,1(⌦⇥]0, T [)3\Co(0, T ;H2(⌦)3),
p 2 Co(0, T ;H2(⌦)). Let {unh(x)}n be the solutions in Voh of
1
k(un+1h, vh) + ⌫(run+1
h ,rvh) = (fn, vh) +1
k(un
hoX0nh , vh), 8vh 2 Voh (88)
where X 0nh is calculated by (77) with un
h replaced by r ⇥ nh and n
h solution of(86), (87).
Then if u is the exact solution of the incompressible Navier-Stokes equations,(1)(4) with u� = 0,we have
(|unh � u(., nk)|20 + ⌫k|r(un
h � u(., nk)|20)12 C[
h2
k+ h+ k] (89)
where the constant C is independent of ⌫, a�ne in ||u||2, ||p||2 and exponential in||u||1,1.
5.6 Other Methods
In practice, all the methods given in chapter 3 for the convection-di↵usionequation can be extended to the Navier-Stokes equations. Paragraph 5 is anexample of such an extension. One can also use the following :
- Lax-Wendro↵ artificial viscosity method : however it cannot be completelyexplicit (same problem as in §4.1) unless one adds pt in the divergence equation(Temam [228], Kawahara [131]);
- the upwinding method by discontinuity ; this could be superimposed withthe other methods of §4.1 and of §4.2. In this way we can obtain methods whichconverge for all ⌫ (cf. Fortin-Thomasset [83], for an example of this type);
- streamline upwinding method (SUPG).
90
5.6.1 SUPG and AIE (Adaptive Implicit/Explicit scheme)
As in paragraph 4.3 in chapter 3 a Petrov-Galerkin variational formulation forthe Navier-Stokes is used with the test functions vh + ⌧uhrvh instead of vh :
(uh,t + uhruh +rph, vh + ⌧uhrvh) + ⌫(ruh,rvh)� ⌫X
l
Z
Tl
⌧uhrvh�uh
= (f, vh + ⌧uhrvh) 8vh 2 V0h(r.uh, qh) = 0 8qh 2 Qh.
Here ⌧ is a parameter of order h, Tl is an element; we have used the simplestSUPG method where the viscosity is added in space only. Johnson [125] rightlysuggest in their error analysis the use of space-time discretisation (see 3.4.3) .
A semi-implicit time discretisation gives the following scheme:
(un+1h � un
h
k+ un
hrun+1h , vh + ⌧un
hrvh) + (rpn+1h , vh) + (rpnh, ⌧u
nhrvh)
+⌫(run+1h ,rvh)� ⌫
X
l
Z
Tl
⌧unhrvh�un
h = (fn+1, vh + ⌧unhrvh) 8vh 2 V0h
(r.un+1h , qh) = 0 8qh 2 Qh.
As noted in Tezduyar [227], it is possible to choose ⌧ so as to have symmetriclinear systems because the non-symmetric part comes from:
(un+1h
k, ⌧un
hrvh) + (unhrun+1
h , vh)
Now this would be zero if ⌧ = k and if unhrun+1
h was equal to r.(unh ⌦ un+1
h ). Sothis suggests the following modified scheme:
Find un+1h � u� 2 V0h and pn+1
h 2 Qh such that
(un+1h � un
h
k+ un
hrun+1h , vh + kun
hrvh) + (r.unh, u
n+1h .vh)
+(rpn+1h , vh) + (rpnh, ku
nhrvh) + ⌫(run+1
h ,rvh)� ⌫X
l
Z
Tl
kunhrvh�un
h
= (fn+1, vh + kunhrvh) 8vh 2 V0h
(r.un+1h , qh) = 0 8qh 2 Qh.
Naturally a Crank-Nicolson O(k2) scheme can be derived in the same way.Similarly a semi-explicit first order scheme could be:
91
(un+1h � un
h
k+ un
hrunh, vh + kun
hrvh) + (rpn+1h , vh) + (rpnh, ku
nhrvh)
+⌫(runh,rvh)� ⌫
X
l
Z
Tl
kunhrvh�un
h = (fn, vh + kunhrvh) 8vh 2 V0h
(r.un+1h , qh) = 0 8qh 2 Qh.
Still following Tezduyar [227] to improve the computing time we can use theexplicit scheme in regions where the local Courant number is large and the implicitscheme when it is small; the decision is made element by element based on thelocal Courant number for convection, |un
h |Tl
k/hl and for di↵usion ⌫k/h2l , and on
a measure of the gradient per element size of unh; here hl is the average size of
element l. So we define n numbers by
�li(c) = max{|un
h|Tl
k
hl� 1, (
U li � ul
i
U⌦i � u⌦
i
)X
j
|@uhi
@xj|� c}
where U li (resp ul
i) is the maximum (resp minimum) of uhi(x) on element Tl andsimilarly for U⌦
i , u⌦i on ⌦ instead of Tl.
The constant c is chosen for each geometry; then if �li(c) > 0 the contribution
to the linear system of the convection terms on element l in the i� th momentumequation is taken explicit (with un
h instead of un+1h ) and otherwise implicit.
Similarly if ⌫k/h2l is less than one, the contribution to the matrix of the linear
system from ⌫RTl
ruh,rvh is taken with uh = unh (explicit) and with un+1
h
otherwise.
5.7 Turbulent flows.
5.7.1 Reynolds’ Stress Tensor.
As we have said in the beginning of the chapter, there are reasons to believe thatwhen t ! 1, the solution of (1)-(4) evolves in a space of dimension proportionalto ⌫�9/4.
For practical applications ⌫ (= 1/Re) is extremely small and so it is necessaryto have a large number of points to capture the limiting solutions in time.
We formulate the following problem (the Reynolds problem) :Let u⌫
w be a (random) solution of
u,t + uru+rp� ⌫�u = 0, r.u = 0 in ⌦⇥]0, T [ (110)
92
u(x, 0) = uo(x) + w(x,!), u|� = u� (111)
where w(x, .) is a random variable having zero average.Let < > be the average operator with respect to the law of u introduced by
w. Can we calculate < u >, < u⌦ u > ... ?This problem corresponds closely to the goals of numerical simulation of the
Navier-Stokes equations at large Reynolds number because, when ⌫ << 1, u isunstable with respect to initial conditions and so the details of the flow cannot bereproduced from one trial to the next. Thus, it is more interesting to find < u >. One may also be interested in < u2 > and ⌫ < |ru|2 > .
Only heuristic solutions of the Reynolds problem are known (see Lesieur [150]for example) but let us do the following reasoning :
If we continue to denote by u the average < u⌫w > and by u0 the di↵erence
u⌫w� < u⌫
w >, then (110) becomes :
u,t + uru+rp� ⌫�u+r.u0 ⌦ u0 = �(u0,t + u0ru+ uru0 +rp0 � ⌫�u0) (112)
r.u0 +r.u = 0 (113)
because uru = r.(u⌦ u) when r.u = 0.If we apply the operator < > to (112) and (113), we see that
u,t + uru+rp� ⌫�u+r. < u0 ⌦ u0 >= 0,r.u = 0 (114)
which is the Reynolds equation and
R =< u0 ⌦ u0 > (115)
is the Reynolds tensor. As it is not possible to find an equation for R as a functionof u, we will use an hypothesis (called a closure assumption) to relate R to ru.
It is quite reasonable to relate R to ru because the turbulent zones are oftenin the strong gradient zones of the flow. But then R(ru) cannot be arbitrarychosen because we must keep (114) invariant under changes of coordinate systems.In fact, it would be absurd to propose an equation for u which is independentupon the reference frame. One can prove (Chacon-Pironneau [50]) that in thiscase the only form possible for R is (see also Speziale [218]).
R = a(|ru+ruT |2)I + b(|ru+ruT |2)(ru+ruT ). (116)
in 2D. In 3D
R = aI + b(ru+ruT ) + c(ru+ruT )2 (117)
where a, b and c are functions of the 2 nontrivial invariants of (ru + ruT ). Sowe get
93
r.R = ra+r.[b(ru+ruT )] +r.[c(ru+ruT )2] (118)
but ra is absorbed by the pressure (we change p to p + a) and so a law of thetype
R = b(ru+ruT ) + c(ru+ruT )2 (119)
has the same e↵ect. In 2D, it is enough to specify a function in one variable b(s)and in 3D two functions in two variables b(s, s0), c(s, s0) where s and s0 are thetwo invariants of ru+ruT .
5.7.2 The Smagorinsky hypothesis
Smagorinsky proposes that b = ch2|ru+ruT |, that is
R = �ch2|ru+ruT |(ru+ruT ), c ⇠= 0.01 (120)
where h is the average mesh size used for (114). This hypothesis is compatiblewith the symmetry and a bidimensional analysis of R ; it is reasonable in 2D butnot su�cient in 3D (Speziale [219]). The fact that h is involved is justified by anergodic hypothesis which amounts to the identification of < > with an averageoperator in a space on a ball of radius h. Numerical experiments show that weobtain satisfying results (Moin-Kim [173]) when we have su�cient points to causeu0 to correspond to the beginning of the inertial range of Kolmogorov.
Remark An implicit scheme with R on the left hand side has the bad e↵ectof coupling all the velocity component. For instance the pressure projectionalgorithm of Chorin is no longer as sequence of scalar PDE. Hence it is better todevise a time iterative algorithm for the Ryenolds equations written in the form:
NS(u, p) = r · ((ch2|ru+ruT |� ⌫)(ru+ruT ))
where ⌫ is close to the average value of ch2|ru+ruT |, and then treat the righthand side in an explicit fashion.
5.7.3 The k � ✏ hypothesis
(Launder-Spalding [143]):
The so-called k � ✏ model was introduced and studied by Launder-Spalding[143], Rodi [200] among others.
Define the kinetic energy of the turbulence k and average rate of dissipationof energy of the turbulence ✏ by:
94
k =1
2< |u0|2 > (121)
✏ =⌫
2< |ru0 +ru0T |2 >; (122)
then R, k, ✏ are modeled by
R =2
3kI � cµ
k2
✏(ru+ruT ), (cµ = 0.09) (123)
k,t + urk � cµ2
k2
✏|ru+ruT |2 �r.(cµ
k2
✏rk) + ✏ = 0 (124)
✏,t + ur✏� c12k
|ru+ruT |2 �r.(c✏k2
✏r✏) + c2
✏2
k= 0 (125)
with c1 = 0.1256, c2 = 1.92, c✏ = 0.07.A rough justification for this set of equations is as follows. First one notes
that k2/✏ has the dimension of a length square so it makes sense to use (123).To obtain an equation for k, (112) is multiplied by u0 and averaged (we recall
that A : B = AijBij) :
1
2< u02 >,t + < u0 ⌦ u0 >: ru+
1
2ur < u02 > +r. < p0u0 > �⌫ < u0�u0 >
+1
2r. < u02u0 >= 0;
That is to say with (123)
k,t � cµk2
✏(ru+ruT ) : ru+ urk� ⌫ < u0�u0 >= � < u0ru02
2> �r. < p0u0 >
The last 3 terms cannot be expressed in terms of u, k and ✏, so they must bemodeled. For the first we use an ergodicity hypothesis and replace an ensembleaverage by a space average on a ball of centre x and radius r, B(x, r):
� < u0�u0 >=< |ru0|2 > +Z
@B(x,r)u0.@u0
@nd�.
By symmetry (quasi-homogeneous turbulence) the boundary integral is small.The second term is modeled by a di↵usion:
< u0ru02
2>⇠= �r. < u0 ⌦ u0 > rk;
95
If u02 and u0 were stochastically independent and if the equation for u02 was linear,it would be exact up to a multiplicative constant, the characteristic time of u0.The third term is treated by a similar argument as the first one so it is conjecturedto be small. So the equation for k is found to be :
k,t + urk � cµ2
k2
✏|ru+ruT |2 �r.(cµ
k2
✏rk) + ✏ = 0 (126)
An intermediate k model
Many have tried to stop there and give a close formula for ✏. On dimensionalground ✏ ⇡ UL U2L�2 = U3/L. Because k is dimensionally U2 we could write
that ✏ = ck32
l where c has no dimension and l has the dimension of a “mixing”
length. Then the turbulent viscosity would be cµlpk. The model is then
k,t + urk � cµ2lpk|ru+ruT |2 �r.(
cµ2lpkrk) + c
k32
l= 0
and
R =2
3kI � cµ
2lpk(ru+ruT ),
The di�culty to choose l is the short coming of this model.
The ✏ equation
To obtain an equation for ✏ one may take the curl of (112),multiply it by r⇥ u0
and use an identity for homogeneous turbulence:
✏ = ⌫ < |r⇥ u0|2 >Letting !0 = r⇥ u0, one obtains:
0 = 2⌫ < !0.(!0,t + (u+ u0)r(! + !0)� (! + !0)r(u+ u0)� ⌫�!0) >
⇠= ✏,t + ur✏+ < u0r⌫!02 > �2⌫ < !0r⇥ (u0 ⇥ !) >
�2⌫(< !0 ⌦ !0 >: ru+r. < (!0 ⌦ !0)u0 >) + 2⌫2 < |r!0|2 >because
r⇥ (u0 ⇥ !) = !ru0 � u0r!.the term < !0r⇥ (u0 ⇥ !) > is neglected for symmetry reasons; < u0r⌫!02 > ismodelled by a di↵usion just as in the k equation; by frame invariance, < !0⌦!0 >should also depend only on ru + ruT , k and ✏ therefore in 2D it can only be
96
proportional to ru+ruT and by a dimension argument it must be proportionalto k. The third term is neglected because it has a small spatial mean and thelast term is modelled by reasons of dimension by a quantity proportional to ✏2/k.Finally one obtains :
✏,t + ur✏� c12k
|ru+ruT |2 �r.(c✏k2
✏r✏) + c2
✏2
k= 0 (127)
The constants are adjusted so that the model makes good predictions for afew simple flows such as turbulence decay behind a grid.
5.7.4 Wall laws
Near walls the k�✏model is not valid because the turbulence is not isotropic. Butfor non-detached mean flow near a wall W experiments show that at a distance�⇤ < � < 100�⇤ from W
u.n = 0, u+ ⌘ u.s
u⇤ = (1
�log �+ + �),
where u⇤ =q⌫|@nu||W , �⇤ =
⌫
u⇤ � ⇡ 5.5
Notice that when @xp = 0 the k � ✏ model contains the log law as a specialstationary solution with
⌫T = �y, k =u⇤2
pcµ, ✏
u⇤3
�y
Numerically these relations should be seen as an implicit relation between u and@nu and inegrated in the variational formulation of the Navier-Stokes equationsin weak form as boundary integrals.
5.7.5 Boundary conditions for the k � ✏ model
Natural boundary condtions could be
k, ✏ given at t = 0 ; k|� = 0, ✏|� = ✏�. (128)
however an attempt can be made to remove the boundary layers from thecomputational domain by considering ”wall conditions” (see Viollet [235] forfurther details)
k|� = u⇤2c� 1
2µ , ✏|� =
u⇤3
K�(129)
97
u.n = 0, ↵u.⌧ + �@u.⌧
@n= � (130)
where K is the Von Karman constant (K = 0.41), � the boundary layer thickness,u⇤ the friction velocity, � = cµ k2/✏, ↵ = cµk2/[✏�(B + log(�/D))] where D is aroughness constant and B is a constant such that u.⌧ matches approximately theviscous sublayer. To compute u⇤, Reichard’s law may be inverted (by Newton’smethod for example) :
u⇤[2.5log(1 + 0.4�u⇤
⌫) + 7.8(1� e�
�u
⇤11⌫ � �u⇤
11⌫e�0.33 �u
⇤⌫ )] = u.⌧
So in reality ↵ and � in (3a) are nonlinear functions of u.⌧, ✏, k.
5.7.6 Positivity of ✏ and k.
For physical and mathematical reasons it is essential that the system yieldspositive values for k and ✏. At least in some cases it is possible to argue thatif the system has a smooth solution for given positive initial data and positiveDirichlet conditions on the boundaries then k and ✏ must stay positive at latertimes.
For this purpose we looks at
✓ =k
✏.
If Dt denotes the total derivative operator, @/@t+ur and E denotes (1/2)|ru+ruT |2, then
Dt✓ =1
✏Dtk � k
✏2Dt✏ = ✓2E(cµ � c1) +
cµ✏r.(
k2
✏rk)� c✏
k
✏2r.(
k2
✏r✏)� 1 + c2
= ✓2E(cµ � c1)� 1 + c2 + cµr.k✓r✓ + 2cµr✓.r(✓
k) + (cµ � c✏)
✓2
kr.k✓r✏
Because cµ < c1 and c2 > 1 , ✓ will stay positive and bounded when there areno di↵usion terms because ✓ is a solution of a stable autonomous ODE along thestreamlines:
Dt✓ = ✓2E(cµ � c1)� 1 + c2
Also ✓ cannot become negative when cµ = c✏ because the moment the minimumof ✓ is zero at (x, t) we will have:
98
r✓ = 0, ✓ = 0
By writing the ✓ equation at this point, we obtain :
✓,t = c2 � 1 > 0
which is impossible because ✓ cannot become negative unless ✓,t 0.Similarly we may rewrite the equation for k in terms of ✓ :
Dtk � cµ2k✓|ru+ruT |2 �r.(cµk✓rk) +
k
✓= 0
Here it is seen that a minimum of k with k = 0 is possible only if Dtk = 0 atthat point, which means that k will not become negative.
Note however that k may have an exponential growth if cµ✓2 |ru+ruT |2 > 2;this will be a valuable criterion to reduce the time step size when it happens.
5.7.7 Numerical Methods.
Let us consider the following model :
u,t + uru+rp�r.[⌫(|ru+ruT |)(ru+ruT )] = 0, r.u = 0 (131)
u.n = 0, au.⌧ + ⌫@u.⌧
@n= b (132)
where a, b are given and ⌫ is an increasing positive function of |ru+ruT | .This problem is well posed ( there is even a uniqueness of solution in 3D under
reasonable hypotheses on ⌫ (cf. Lions [153])).The variational formulation can be written in the space of functions having
zero divergence and having normal trace zero :
(u,t, v) + (uru, v) +1
2(⌫(|ru+ruT |)(ru+ruT ),rv +rvT ) (133)
+Z
�[au.v � bv]d� = 0
8v 2 Jon(⌦); u 2 Jon(⌦) = {v 2 H1(⌦)3 : r.u = 0, v.n|� = 0} (134)
By discretising the total derivative, we can consider a semi-implicit scheme,
1
k(un+1
h � unhoX
nh , vh) +
1
2(⌫nh (run+1
h + (run+1h )T ),rvh + (rvh)
T ) (135)
99
+Z
�(aun
h � b)vhd� = 0, 8vh 2 Jonh
un+1h 2 Jonh = {vh 2 Vh : (r.vh, qh) = 0 8qh 2 Qh; vh.nh|� = 0} (136)
where Vh and Qh are as in chapter 4 and where :
⌫nh = ⌫(|runh + (run
h)T |) (137)
There is a di�culty with Jonh and the choice of the approximated normal nh,especially if ⌦ has corners.
We can also replace Jonh by
J 0onh = {vh 2 Vh : (vh,rqh) = 0 8qh 2 Qh} (138)
because
0 = (u,rq) = �(r.u, q) +Z
�u.nqd�, 8q ) r.u = 0, u.n|� = 0. (139)
With J 0onh the slip boundary conditions are satisfied in a weak sense only but the
normal nh does not now appear .The techniques developed for Navier-Stokes equations can be adapted to this
framework, in particular the solution of the linear system (134) can be carriedout with the conjugate gradient algorithm developed in chapter 4. However wenote that the matrices would have to be reconstructed at each iteration because⌫ depends on n.
To solve (114), (123) (the equations k� ✏), we can use the same method ; weadd to (133)-(134) (k is replaced by q)
(qn+1h � qnhoX
nh , wh) + kcµ(
qn2
h
✏nhrqn+1
h ,rwh) (141)
+(Z (n+1)k
nk[✏nh � cµ
qn2
h
2✏nh|run+1
h +run+1h |2](X(t))dt, wh) = 0, 8wh 2 Woh
(✏n+1h � ✏nhoX
nh , wh) + kc✏(
qn2
h
✏nhr✏n+1
h ,rwnh) (142)
+(Z (n+1)k
nk�[
c12qnh |run+1
h +run+1Th |2 + c2
✏nhqnh
](X(t))dt, wh) = 0, 8wh 2 Woh
The integrals from nk to (n+ 1)k are carried out along the streamlines in orderto stabilize the numerical method (Goussebaile-Jacomy [99]).
100
So, at each iteration, we must- solve a Reichard’s law at each point on the walls,- solve (134),- solve the linear systems (141)-(142).The algorithm is not very stable and converges slowly in some cases but it
may be modified as follows.
As in Goussebaile et al.[99] the equations for q � ✏ are solved by a multistepalgorithm involving one step of convection and one step of di↵usion. However inthis case the convection step is performed on q, ✓ rather than on q, ✏.
The equation for q is integrated as follows:
(qn+1h , wh)� (qnhoX
nh , wh) + (qn+1
h
Z (n+1)k
nk(cµ2
qnh✏nh
|runh +run
h|2 �✏nhqnh
), wh) (143)
+kcµ(qn2h✏nh
rqn+1h ,rwh) = 0 8wh 2 Qoh = {wh 2 Qh : wh|� = 0}
qh � k�h 2 Qoh (144)
But the equation for ✏ is treated in two steps via a convection of ✓ = q/✏ thatdoes not include the viscous terms
(✓n+ 1
2h , wh) + (✓
n+ 12
h
Z (n+1)k
nk
1
2✓nh |run
h +runTh |2(c1 � cµ), wh) (145)
= (✓nhoXnh , wh) + (c2 � 1, wh)k
where ✓nh = qnh/✏nh. Then ✏
n+1/2h is found as
✏n+ 1
2h =
qn+1h
✓n+h12
(146)
and a di↵usion step can be applied to find ✏n+1h :
(✏n+1h , wh) + kc✏(
qn2h✏nh
r✏n+1h ,rwh) = (✏
n+ 12
h , wh) (147)
8wh 2 Qoh; ✏n+1h � ✏�h 2 Qoh
Notice that this scheme cannot produce negative values for qn+1h and ✏n+1
h when
Z
]x,Xn
h
(x)[(cµ2✓nh |run
h +runh|2 �
1
✓nh) > �1 (148)
because c1 > cµ and c2 > 1 so (145) generates only positive ✓ while in (143) thecoe�cient of qn+1
h is positive. Note that (148) is a stability condition when theproduction terms are greater than the dissipation terms.
Chapter 6
Finite Volumes for Euler,Navier-Stokes and Saint Venant’sequations
6.1 Compressible Euler Equations
6.1.1 Position of the problem
The general equations of a perfect fluid in 3 dimensions can be written as (cf.(1.2), (1.7), (1.12))
W,t +r.F (W ) = 0 (1)
and, in relation to (1.12)
E = ⇢(e+1
2|u|2)
and
W = [⇢, ⇢u1, ⇢u2, ⇢u3, E]T (2)
F (W ) = [F1(W ), F2(W ), F3(W )]T (3)
Fi(W ) = [⇢ui, ⇢u1ui + �1ip, ⇢u2ui + �2ip, ⇢u3ui + �3ip, ui(E + p)]T (4)
p = (� � 1)(E � 1
2⇢|u|2) (5)
The determination of a set of Dirichlet boundary conditions for (1) to be welldefined is a di�cult problem. Evidently we need initial conditions
W (x, 0) = W o(x) 8x 2 ⌦ (6)
101
102
To determine the boundary conditions, we write (1) as :
W,t +X
Ai(W )@W
@xi= 0 (7)
where Ai(W ) is the 5 ⇥ 5 matrix whose elements are the derivative of Fi withrespect to Wj.
Let ni(x) be a component of the external normal to � at x. One can showthat
B(W,n) =X
Ai(W )ni (8)
is diagonalizable with eigenvalues
�1(n) = u.n� c, �2(n) = �3(n) = �4(n) = u.n, �5(n) = u.n+ c (9)
where c = (�p/⇢)1/2 is the velocity of sound. Finally, by analogy with the linearproblem, we see that it is necessary to give Dirichlet conditions at each point xof � with negative eigenvalues. We will then have 0, 1, 4 or 5 conditions on thecomponents of W according to di↵erent cases. The important cases are:
• supersonic entering flow ( u.n < 0, |u.n| > c) : 5 conditions for example, ⇢,⇢ui, p
• supersonic exit flow : 0 conditions.
• subsonic entering flow : 4 conditions, independent combinations of W.li
where li is the left eigenvector associated to the eigenvalue �i < 0.
• subsonic exit flow : 1 condition (in general, on pressure)
• flow slipping at the surface (u.n = 0): 1 condition, i.e.: u.n = 0.
6.1.2 Bidimensional test problems
a) Flow through a canal with an obstacle in the form of an arc of circle (Rizzi-Viviand[199])
The height of the hump is 8% of its length. The flow is uniform and subsonicat the entrance of the canal, such that the Mach number is 0.85. We have slippingflow along the horizontal boundary and along the arc (u.n = 0); the quality ofthe result can be seen, for example, from the distribution of entropy created bythe shocks.
103
b) Flow in a backward step canal (Woodward-Colella[238])
This flow is supersonic at the exit and we give only conditions at the entrancep = 1, ⇢ = 1.4, u = (3, 0). The quality of the results can be seen in the positionof the shocks and of the L contact discontinuity lines.
6.2 Finite Volumes and Upwinding by disconti-nuity
Upwinding through the discontinuities of F (Wh) at the inter-element boundaries,can be done when Wh is a discontinuous approximation of W , constant on eachelement for example, or even though all Wh continuous, we know how to associatea discontinuous value at right and at left, at the inter-element boundary.
If Wh is continuous piecewise linear on each triangle (tetraedra in 3D), to eachvertex qi we can associate a volume �i by dividing each triangles into subtrianglesby the 3 medians and keeps the triangular subtriangles which contains qi.
a) General Framework (Dervieux [64], Fezoui [78], Stou✏et et al. [222]).As above, for a given triangulation, we associate to each vertex qi a cell �i
obtained by dividing triangles (tetraedra) by the medians (by median planes).Thus we could associate to each piecewise continuous function in a triangulationa function P o (piecewise constant ) in �i by the formula
W ph |�i =
1
|�i|Z
�i
Whdx (22)
By multiplying (1) by a characteristic function of �i and by integration (Petrov-Galerkin weak formulation) we obtain, after an explicit time discretisation, thefollowing scheme :
W n+1h (qi) = W n
h (qi) +
k
|�i|Z
@�i
Fd(Wph ).n 8i (23)
On �i \ �, we take Fd(W ) = F (W ) and we take into account the knowncomponents of W� ; and elsewhere Fd(W ) is a piecewise constant approximationof F (W ) verifying
Z
@�i
Fd(Wph ).n =
X
j 6=i
�(W ph|�i
,W ph|�j
)Z
@�i\�j
n (24)
where � will be defined as a function of F (W ph|�i
) and F (W ph|�j
); �(u, v) is thenumerical flux function chosen according to the qualities sought for the scheme
104
(robustness precision ease of programming). In all cases this function shouldsatisfy the consistency relation :
�(V, V ) = F (V ), for all V . (25)
b) Definition of the flux � :Let B(W,n) 2 R5⇥5 (4⇥ 4 in 2D) be such that
F (W ).n = B(W,n).W 8W 8n (26)
Note that B is the same in (8) because F is homogeneous of degree 1 in W(F (�W ) = �F (W )) and so B is nothing but F 0
i (W )ni. As we have seen that Bis diagonalizable, there exists T 2 R5⇥5 such that
B = T�1⇤T (27)
where ⇤ is the diagonal matrix of eigenvalues.We denote
⇤± = diag(±max(±�i, 0)), B± = T�1⇤±T (28)
|B| = B+ � B�, B = B+ +B� (29)
We can choose for � one of the following formulae :
�SW (V i, V j) = B+(V i)V i +B�(V j)V j (Steger-Warming) (30)
�V S(V i, V j) = B+(V i + V j
2)V i +B�(
V i + V j
2)V j (Vijayasundaram) (31)
�V L(V i, V j) =1
2[F (V i) + F (V j) + |B(
V i + V j
2)|(V j � V i)] (Van Leer) (32)
�OS(V i, V j) =1
2[F (V i) + F (V j)�
Z V j
Vi
|B(W )|dN ] (0sher) (33)
the guiding idea being to get �l(V i, V j) ⇠= F (V i)l if �l is positive and F (V j)l if�l < 0.
So �SW , for instance, can be rewritten as follows :
�SW (V i, V j) =1
2[F (V i) + F (V j)] +
1
2[|B(V i)|V i � |B(V j)|V j],
because
F (V i) + F (V j) = (B+(V i) + B�(V i))V i + (B+(V j) + B�(V j))V j;
105
The first term, if alone, would yield an upwind approximation. The second, aftersummation on all the neighbors of i, is an artificial viscosity term. The VanLeer and Osher schemes rely also on such an identification ; the flux of Osher isbuilt from an integral so that it is C1 continuous ; the path in R5 from V i to V j
is chosen in a precise manner along the characteristics so as to capture exactlysingularities like the sonic points.
c) Integration in time.The previous schemes are stable up to CFL of order 1 (c|w|k/h < 1 where c
depends on the geometry); if one wants only the stationary solution, the schemecan be speeded up by a preconditioning in front of @W/@t ; these schemes arequite robust ; 3D flows at up to Mach 20 can be computed ; but they are notprecise. Schemes of order 2 are being studied.
6.3 Compressible Navier-Stokes Equations
6.3.1 Generalities
The equations (1.2), (1.7), (1.12) can also be written as a vectorial system in W .With the notations (2)-(5), we have
W,t +r.F (W )�r.K(W,rW ) = 0 (44)
where K(W,rW ) is a linear 2nd order tensor in rW such that
Ki,1 = 0, K.,2,3,4 = ⌘ru+ (⇣ +⌘
3)Ir.u, (45)
K.,5 = ⌘(ru+ruT )u+ (⇣ � 2
3⌘)ur.u+ r(
E
⇢� |u|2
2) (46)
The term r.K(W,rW ) is seemingly a di↵usion term if ⇢, ✓ > 0 since itcorresponds to ⌘�u + (⌘/3 + ⇣)r(r.u) in the 2nd, 3rd and 4th equations of (44)and to (Cv/⇢)�✓ + 1/⇢Cv[|r.u|2(⇣ � 2⌘/3)+ |ru + ruT |⌘/2] in the last one(cf. (1.13)):
�Z
R3Wr.K(W,rW ) = �
Z
R3[⌘�u+ (
⌘
3+ ⇣)r(r.u)]u (47)
�Z
R3(Cv
✓
⇢�✓ +
✓
⇢Cv[|r.u|2(⇣ � 2
⌘
3) + |ru+ruT |⌘
2])
�Z
R3(⌘|ru|2 + (
⌘
3+ ⇣)|r.u|2) +
Cv
⇢max
Z
R3|r✓|2
106
+✓min
⇢maxCv
Z
R3[|r.u|2(⇣ � 2
⌘
3) +
⌘
2|ru+ruT |2] � 0
So we can think that the explicit schemes in step 1 to integrate (1) can be appliedto (44) if the stability condition on the time step k is modified :
k Cmin[h
|w| ,h2
|K| ] (48)
where K is a function of , ⌘, ⇣. This modification is favorable to the schemesobtained by artificial viscosity but still when the boundary conditions are di↵erentboundary layers arise boundary layers in the neighborhood of the walls.
The boundary conditions can be treated in the strong sense (they appear inthe variational space Voh instead of Vh) or in the weak sense (they appear as aboundary term in the variational formulation).
Flow around a NACA0012 airfoilA test problem is given in Bristeau et al.[43] which concerns a 2D flow around
a wing profile NACA0012. Taking the origin of the reference frame at the leadingedge the equation of the upper surface of the profile is :
y = 0.17735px� 0.075597x� 0.212836x2 + 0.17363x3 � 0.06254x4, 0 x 1
The boundary conditions at infinity are such that the Mach number is 0.85,the Reynolds number (u⇢/⌘) is 500 and the angle of attack is 00 or 100. Thetemperatures at infinity and on the surface are given.
6.4 Saint-Venant’s Shallow Water Equations
6.4.1 Generalities
Let us go back to the incompressible Navier-Stokes equations with gravity(g = �9.81)
u,t +r.u⌦ u+rp� ⌫�u = ge3 r.u = 0 (56)
and assume that the domain occupied by the fluid has a small thickness, whichis true in the case of lakes and seas .
Let zs(x1, x2, t) be the height of the surface and zf (x1, x2, t) the height of thebed. The continuity equation integrated in x3 = z gives
107
Z zs
zf
r.udz = u3(zs)� u3(zf ) +r12.(Z z
s
zf
udz) = 0 (57)
but by the definition of u3 we have dzs/dt = u3(zs) so, taking into account theno-slip condition at the bottom and defining v, z by
Z zs
zf
udz = (zs � zf )v(x1, x2, t), z = zs � zf (58)
one can rewrite (57) as
z,t +r.(zv) = 0 (59)
In (56)(a) we neglect all the terms in u3, so the third equation gives
@p
@z= g (60)
By putting p calculated from (60) in the two first equations of (56)(a) integratedin x3 we see
(zv),t +r.(zv ⌦ v) + gzrz � ⌫�(zv) ⇠= 0 (61)
The system (59), (61) constitutes the shallow water equations of Saint-Venant(for more details, see Benque et al. [22], for example). For large regions of water(seas), the Coriolis forces, proportional to !⇥ v where ! is the rotation vector ofthe earth must be added. Other source terms, f , in the right hand side of (61)could come from
- the modeling of the wind e↵ect (f constant)- the modeling of the friction at the bottom zf (f = c|v|v/z).If we change scales in (59), (61) as in (1.39),we obtain
z0,t0 +r0.(z0v0) = 0 (62)
(z0v0),t0 +r0.(z0v0 ⌦ v0) + F�2z0r0z0 �R�1�0(z0v0) = 0 (63)
with R = |v|L/⌫ and the Froude number
F =|v|pgz
(64)
If we expand (61) and divide by z,we get
v,t + vrv + grz � ⌫z�1�(zv) = 0 (65)
If F << 1 and R << 1, then grz dominates the convection term vrv and thedi↵usion term ⌫z�1�zv ; we can approximate (65) (59) by,
108
z,t +r.(zv) = 0 v,t + grz = 0 (66)
which is a nonlinear hyperbolic system with a propagation velocity equal topgz.
If R >> 1 so that we can neglect the viscosity term, we remark that (59),(61) is similar to the Euler equations with � = 2, ⇢ = z and an adiabaticapproximation p/⇢� = constant ; on the other hand if in (65) we set u = zvthen it becomes similar to the incompressible Navier-Stokes equations with anartificial compressibility such as studied by Temam [228]:
u,t +r.(z�1u⌦ u) +g
2rz2 � ⌫�(u) = 0 z,t +r.u = 0;
this problem is well posed with initial data on u, z and Dirichlet boundary dataon u only.
But if we make the following approximation:
z�1�(zv) ⇠= �v,
then (59)(61) becomes similar to the compressible Navier-Stokes equations with� = 1 :
v,t + vrv + grz � ⌫�v = 0
z,t +r.(zv) = 0;
An existence theorem of the type found by Matsumura-Nishida [169] may beobtained for these equations ; this means that the problem (59), (61) is wellposed with the following boundary conditions :
z, v given at t = 0 (67a)
v|� given � and z given on all points of � where v.n < 0 (67b)
with the condition that z stays strictly positive. This fact is confirmed by ananalysis of the following problem (Pironneau et al [192]):
grz � ⌫�v = 0 r.(zv) = 0;
with (67b). This reminds us that certain seemingly innocent approximations havea dramatic e↵ect on the mathematical properties of such systems.
Boundary conditions for these shallow water equations is a serious problem;here are some di�cult examples found in practical problems:
- on the banks where z ! 0� in deep sea where one needs non reflecting boundary conditions if the
computational domain is not to be the wholesurface of the earth.
109
- if the water level goes down and islands appear ( z ⌘ 0).
Test Problem :A simple non stationary test problem is to study a wave which is axisymmetric
and Gaussian in form with 2m height in a square of 5m depth and 10m sidereturning to rest.
6.4.2 Numerical scheme in height-velocity formulation
When the Froude number is small, we shall use the methods applicable to in-compressible Navier-Stokes equations rather than the methods related to Euler’sequations.
We shall neglect the product term rz rv (small compared to vrv) in thedevelopment of �(zv) in the formulation (59), (65), that is we shall consider thesystem
z,t + vrz + zr.v = 0 (68)
v,t + vrv + grz � ⌫�v = 0 (69)
z(0) = zo, v(0) = vo in ⌦; v|� = v�, z|⌃ = z� (70)
For clarity, we assume that v� = 0. Recall that ⌃ is that part of � where the fluxenters (v.n < 0). If D/Dt is the total derivative, (68)-(69) can also be written
Dz
Dt+ zr.v = 0 (71)
Dv
Dt+ grz � ⌫�v = 0. (72)
With the notation of chapter 3, we can discretise the system with the semi-implicitEuler scheme
1
k(zm+1 � zmoXm) + zmr.vm+1 = 0 (73)
1
k(vm+1 � vmoXm) + grzm+1 � ⌫�vm+1 = 0 (74)
For spatial discretisation, we shall use the methods given in chapter 4 for theStokes problem : we choose Qh
⇠= L2(⌦) and Voh⇠= H1
o (⌦)n and we find
[zm+1h , vm+1
h ] solution of
1
k(zm+1
h ,qhzmh
) + (r.vm+1h , qh) =
1
k(zmh oXm
h ,qhzmh
) 8qh 2 Qh (75)
110
1
k(vm+1
h , wh) + ⌫(rvm+1h ,rwh) + g(rzm+1
h , wh) =1
k(vmh oX
mh , wh) 8wh 2 Voh
(76)We recall that Xm
h (x) is an approximation of X(mk, x), which is a solution of
dX
d⌧= vmh (X(⌧), ⌧), X((m+ 1)k) = x, (77)
thus the boundary condition on zh|⌃ appears in (75) in the calculation ofzmh oXm
h (x), x 2 ⌃.At each iteration in time, we have to solve a linear system of the type
✓A BBT �D
◆✓VZ
◆=
✓FG
◆(78)
where
Dij =1
k
Z
⌦
qiqj
zmh, Bij = (rqi, wj), Gj = �1
k(zmh oXm
h ,qj
zmh)
Fj =1
g(vmh oX
mh , wj), Aij =
1
kg(wi, wj) +
⌫
g(rwi,rwj) (79)
As with the Stokes problem, the matrix of the linear systems is symmetric buthere it is nonsingular whatever may be the chosen element {Vh, Qh}. In fact, byusing (76), to eliminate vm+1
h , we find an equation for zm+1h :
(BTA�1B +D)Z = BTA�1F �G (80)
the matrix of this linear system is always positive definite (cf. (79)). It canbe solved by the conjugate gradient method exactly as in the Stokes problem. Apreconditioner can also be constructed in the same manner ; Goutal [100] suggests:
C =D
kg��h (81)
where D is given in (79) and ��h is a Laplacian matrix with Neumann condition(this preconditioning corresponds to a discretisation of the operator on z in thecontinuous case when ⌫ = 0).
Stability and convergence are open problems but the numerical performanceof the method is good[13].
6.4.3 A numerical scheme in height-flux formulation
We put D = zv so that (59)-(61) can be rewritten as
111
z,t +r.D = 0 (82)
D,t +r.(1
zD ⌦D) + gzrz � ⌫�D = 0 (83)
Let us take as a model problem the case where
z(0) = zo, D(0) = Do in ⌦ (84)
D = D� on �⇥]0, T [ (85)
In [13] the following scheme is studied : with the same finite element space V0h
and Q0h, the functions in the space Qh which are zero on �, one solves
1
k(zm+1
h � zmh , qh) + (r.Dm+1h , qh) = 0 8qh 2 Qh (86)
1
kg(Dm+1
h
zmh� vmh oX
mh , wh) + (rzm+1
h , wh) +⌫
g(rDm+1
h ,rwh
zmh) (87)
+1
g(vmh r.vmh , wh) = 0 8wh 2 Voh
where vm = Dmh /z
mh and where Xm
h is as in the previous paragraph, that is, it iscalculated with vh : Xm
h (x) ⇠= x� vmh (x)k .Each iteration requires the resolution of a linear system of the type (78) but
with
Dij =1
k(qi, qj) Aij =
1
kg
Z
⌦[wiwj
zmh+ ⌫rwirwj
zmh] (88)
We note that A is no longer symmetric except when ⌫ = 0. The uniqueness of thesolution of (86)-(87) is no longer guaranteed except when ⌫ << 1 which makes itsymmetric again. When ⌫ 6= 0, we can also use the following approximation
(rDm+1h
zmh,rwh) ⇠=
Z
⌦
1
zmhrDm+1
h rwh �Z
⌦
Dmh
(zmh )2rzmh rwh (89)
Another method can be devised by working with z⇤ = z2 instead of z and byreplacing (zm+1
h � zmh , qh) in (86) by (z⇤m+1h � z⇤mh , qh/(2zmh )). The convergence of
this algorithm is also an open problem.
112
6.4.4 Comparison of the two schemes
The main di↵erence between the height-velocity and height-flux formulations isin the treatment of the convection term in the continuity equation and in theboundary conditions. We expect the first formulation to be more stable for flowswith high velocity. The other factors which should be considered in selecting amethod are :
- the boundary conditions,- the choice of variable to be conserved- the presence of shocks (torrential regime, ⌫ = 0) ; (86) is in conservative
form whereas (75) is not.
6.5 Discontinuous - Galerkin Methods
DG as it called nowadays is a more systematic way to control the upwindingby discontinuities. The simple schemes studied in the previous section are toodi↵usive for real Navier-Stokes problem of CFD; thus many people are currentlyadapting DG to their solvers. As it is rather complex the best is to consider anexample: Assume r · a = 0, let u be the solution of
µu+r · (au) = f in ⌦, u|@⌦� = g
Lesaint’s method
The oldest method dates from the seventies: find u 2 Vh such that
(µu+ aru, w) +X
T
Z
@T[u]w(a · n)� = (f, w), 8w 2 Vh
with the convention that [u] is the value of u outside T minus the value inside,that n points outside T and that v� = �min(v, 0). In the above g must be usedwhenever u outside the domain is required. Later we will also use v to mean theaverage of v outside T and v inside T .
Typically Vh is the set of piecewise polynomial functions discontinuous on theedges of the triangulation.
Remark : A penalty can be added as in SUPG
(µu+ aru, w + ✓arw) +X
T
Z
@T[u]w(a · n)� = (f, w + ✓arw)
113
Two Classical DG formulations
Dual formulation
(µu+ arvu, w) +X
T
Z
@T /2@⌦(↵|a · n|� 1
2a · n)[u]w =
X
T
Z
@T2@⌦(a · n)�gw+ (f, w)
Notice that ↵ = 12 gives the previous scheme.
Primal formulation (g = 0)
(u, µw � arw) +X
T
Z
@T /2@⌦(↵|a · n|[u] + a · nu)w +
X
T
Z
@T2@⌦(a · n)+uw = (f, w)
Theorem Assume u, w 2 Vh the space of piecewise discontinuous polynomialsof degree p. Then
(ku� uhk20 +X
@T
|a · n|[uh]2 +
X
T
hT |a ·r(u� uh)|2) 12 Chp+ 1
2
Remark The method can be extended to convection di↵usion problem bywriting it as a system of 2 first order equations.
6.5.1 SUPG mixed with Lesaint’s method
Assume r · u = 0
a�+r · (u�) = f
discretized by, w+ is the value on the triangle, w� the value outside T and @T�
the part on which u · n < 0:
(a�+ ur�, w + ✓urw)�Z
@T�[�]w+u · n = (f, w + ✓urw)
Theorem
k�ku < C(|f |20 + |�2u · n|0,��)1/2
where
k�k2u = (|a1/2�|20 +1
2
XZ
@T�[�]2|u · n|+ 1
2
Z
�+�2u · n)1/2
114
6.5.2 Di↵usion Problems
�r · (⌫ru) + µu = f plus Neumann = g|�N
and Dirichlet on �D
Set
b(u, w) = ⌫(ru,rw) + µ(u, w)J(u, w) =
X
@T
⌫ru · n[w] + (⌫ru · nw)�D
L(w) = (f, w) + (g, w)�N
m±(u, w) = b(u, w)� J(u, w)± J(w, u)L± = L(w)± (⌫rw · n, uD)�
D
J✓(u, w) =X
@T /2�✓[u][w] + (u, w)�
D
(6.1)
Method 1. Solve
b(u, w)� J(u, w) = L(w) 8w 2 H2(T )
Method 2. (Delves-Hall[259] Solve
m�(u, w) = L�(w)
As m� may not be > 0 a penalty is added
Method 3 (Douglas-Dupont[257], Wheeler[262], Arnold[255]))
m(u, w) + J✓(u, w) = L�(w) + ✓(uD, w)�D
with ✓ > C/h the bilinear form is coercive and continuous and optimal in H1.
Method 4. (Oden[260], Riviere[261])
m+(u, w) = L+(w) 8w 2 H2(T )
The bilinear form is not symmetric but it is continuous coercive and the methodis optimal in H1 norm.
Penalty can also be added (Riviere[261]).
Bibliography
[1] O. Pironneau: Numerical Methods for Fluids. Wiley (1990)
[2] B. Mohammadi and O. Pironneau. (1994). Analysis of the k-epsilon turbu-lence model, Wiley.
[3] B.E. Launder and D.B. Spalding. (1972). Mathematical models of turbulence,Academic Press.
[4] D. Vandromme. (1983). Contribution a la modelisation et la predictiond’ecoulements turbulents a masse volumique variable, Thesis, University ofLille.
[5] G. Comte-Bellot and S. Corsin. (1971) Simple eulerian time-correlation offull and narrow-band velocity signals in grid-generated isotropic turbulence, J.Fluid Mech., 48, 273-337.
[6] S. Thangam. (1991). Analysis of two-equation turbulence models for recir-culating flows, ICASE report 91-61.
[7] B. Mohammadi. (1992). Complex turbulent compressible flows computationwith a two-layer approach, Int. J. Num. Meth. for Fluids, 15, 747-771.
[8] J. Cousteix. (1990). Turbulence et couche limite, Cepadues publisher.
[9] E.R. Van Driest. (1951). Turbulent boundary layers in compressible fluids, J.of Aeronautics Science, 18-145.
[10] J. Steger and R.F. Warming. (1983). Flux vector splitting for the inviscidgas dynamic with applications to finite-di↵erence methods, J. Comp. Phys. 40,263-293.
[11] B. Mohammadi and G. Puigt. (2000). Generalized wall functions for high-speed separated flows over adiabatic and isothermal walls, Int. J. Comp. FluidDynamics, 14, 20-41.
115
116
[12] B. Mohammadi, G. Puigt. (2006). Wall Functions in Computational FluidMechanics, Computers & Fluids, Vol. 35, num. 10.
[13] F. Angrand: Viscous perturbation for the compressible Euler equations,application to the numerical simulation of compressible viscous flows. M.O.Bristeau et al. ed. Numerical simulation of compressible Navier-Stokes flows .Notes on Numerical fluid mechanics, 18 , Vieweg, 1987.
[14] F. Angrand, A. Dervieux: Some explicit triangular finite elements schemesfor the Euler equations. J. for Num. Meth. in Fluids , 4 , 749-764, 1984.
[15] H. Akay, A. Ecer: Compressible viscous flows with Clebsch variables. inNumerical methods in laminar and turbulent flows . C. Taylor, W. Habashi, M.Hafez ed. Pineridge press 1989-2001, 1987.
[16] J.D. Anderson, Jr.: Modern compressible flow with historical perspective.McGraw Hill. 1982. see also Fundamental Aerodynamics Mc-Graw Hill 1984.
[17] D. Arnold,F. Brezzi, M. Fortin: A stable finite element for the Stokesequations. Calcolo 21(4 ) 337-344 ,1984.
[18] O. Axelson, V. Barker: Finite element solution of boundary value problems.Academic Press 1984.
[19] K. Baba, M. Tabata: On a conservative upwind finite element schem forconvection di↵usion equations. RAIRO numer Anal . 15 ,1, 3-25, 1981.
[20] I Babuska, W. Rheinboldt: Error estimates for adaptive finite elementcomputations. SIAM J. Numer Comp . 15. 1978.
[21] I. Babuska: the p and h�p version of the finite element method: the state ofthe art. in in Finite element, theory and application . D. Dwoyer, M Hussaini,R. Voigt eds. Springer ICASE/NASA LaRC series. 199-239 1988.
[22] G.K. Bachelor: An Introduction to Fluid Dynamics. Cambridge UniversityPress, 1970 .
[23] A. Baker: Finite element computational fluid mechanics . McGraw-Hill,1985.
[24] R. Bank: PTLMG User’s guide. Dept of Math. UCSD tech. report. Jan 1988.
[25] R. Bank, T. Dupont, H. Yserentant: The Hierarchical basis multigridmethod. Konrad Zuse Zentrum Preprint SC-87-1. 1987.
117
[26] R. Bank, L.R. Scott: On the conditioning of Finite Element equations withhighly refined meshes. Math Dept. PennState research report 87013. 1987.
[27] R. Bank, T. Dupont, H. Yserenlant : The Hierarchichal multigridmethod.UCLA math preprint SC87-2, 1987.
[28] C. Bardos: On the two dimensional Euler equations for incompressible fluids.J. Math. Anal. Appl ., 40 , 769-790, 1972.
[29] T.J. Beale, A. Majda: Rate of convergence for viscous splitting of the Navier-Stokes equations. Math. Comp . 37 243-260, 1981.
[30] C. Begue, C. Conca, F. Murat, O. Pironneau: A nouveau sur les equationsde Navier-Stokes avec conditions aux limites sur la pression. Note CRAS t 304Serie I 1 .,23-28 , 1987
[31] C. Begue, C. Conca, F. Murat, O. Pironneau: Equations de Navier-Stokesavec conditions aux limites sur la pression. Nonlinear Partial Di↵erentialequations and their applications. College de France Seminar 9 , (H. Brezisand J.L. Lions ed) Pitman 1988.
[32] J.P. Benque, O. Daubert, J. Goussebaile, H. Haugel: Splitting up techniquesfor computations of industrial flows. InVistas in applied mathematics . A.V.Balakrishnan ed. Optimization Software inc. Springer, 1986.
[33] J.P. Benque, A. Haugel, P.L. Viollet,: Engineering applications of hydraulicsII . Pitman, 1982.
[34] J.P. Benque - B. Ibler - A. Keraimsi - G. Labadie: A finite element methodfor the Navier-Stokes eq. In Norries ed. 1110-1120, Pentech Press. 1980.
[35] J.P. Benque, B. Ibler, A. Keramsi, G. Labadie: A new finite element methodfor Navier-Stokes equations coupled with a temperature equation. In T. Kawai,ed. Proc. 4th Int. Symp. on Finite elements in flow problems . North-Holland295-301, 1982.
[36] J.P. Benque, G. Labadie, B. Latteux: Une methode d’elements finis pour lesequations de Saint-Venant, 2nd Conf. On numerical methods in laminar andturbulent flows. Venice, 1981.
[37] M. Bercovier : Perturbation of mixed variational problems. RAIRO serierouge 12(3 ) 211-236 ,1978.
[38] M. Bercovier - O. Pironneau: Error Estimates for Finite Element solutionof the Stokes problem in the primitive Variables. Numer. Math . 33, 211-224,1979.
118
[39] M. Bercovier, O. Pironneau, V. Sastri: Finite elements and characteristics forsome parabolic-hyperbolic problems:Appl. Math. Modelling , 7 , 89-96, 1983.
[40] A. Bermudez, J. Durani: La methode des caracteristiques pour les problemesde convection-di↵usion stationnaires. MA2N RAIRO 21 7-26, 1987.
[41] P. Berge, Y. Pomeau, Ch. Vidal: De l’ordre dans le chaos . Hermann 1984.
[42] C. Bernardi. These de doctorat 3eme cycle, Universite Paris 6, 1979.
[43] C. Bernardi, G. Raugel: A conforming finite element method for the time-dependant Navier-Stokes equations. SIAM J. Numer. Anal. 22 , 455-473, 1985.
[44] S. Boivin: A numerical method for solving the compressible Navier-Stokesequations. (to appear in IMPACT 1989).
[45] J. Boland, R. Nicolaides: Stability of finite elements under divergenceconstraints.SIAM J. Numer. Anal . 20 ,4, 722-731. 1983.
[46] J. Boris, D.L. Book: Flux corrected transport. J. Comp. Phys. 20 397-431.1976.
[47] A. Brandt: Multi-level adaptive solution to boundary value problems. Math.of comp.31, 333-391, 1977.
[48] C. Brebbia: The Boundary Element Method for Engineers. Pentech, 1978.
[49] Y. Brenier: The transport-collapse algorithm, SIAM J. Numer Anal . 21p1013, 1984
[50] F. Brezzi, J. Douglas: Stabilized mixed methods for the Stokes problem. (toappear).
[51] F. Brezzi: On the existence, uniqueness and approximation of saddle-pointproblems arising from Lagrange multipliers. RAIRO Anal Numer. 8 , 129-151,1974.
[52] M.O. Bristeau, A. Dervieux: Computation of directional derivatives at thenodes: §2.4 in F. Thomasset: Implementation of finite element methods forNavier-Stokes equations , Springer series in comp. Physics, 48-50, 1981.
[53] M.O. Bristeau, R. Glowinski, B. Mantel, J. Periaux, C. Pouletty: Solution ofthe compressible Navier-Stokes equations by least-squares and finite elements.In Bristeau et al. ed:Numerical simulation of compressible Navier-Stokes flows. Notes on Numerical fluid mechanics, 18 , Vieweg, 1987.
119
[54] M.O. Bristeau , R. Glowinski, B. Mantel, J. Periaux, G. Roge: Adaptivefinite element methods for three dimensional compressible viscous flow simula-tion in aerospace engineering. Proc. 11th Int. Conf. on Numerical methods influid mechanics. Williamsburg, Virginia (1988) ( Springer , to appear )
[55] M.O. Bristeau, R. Glowinski, J. Periaux, H. Viviand: Numerical simulationof compressible Navier-Stokes flows . Notes on Numerical fluid mechanics, 18, Vieweg, 1987.
[56] M.O. Bristeau, R. Glowinski, J. Periaux, O. Pironneau, P. Perrier: On thenumerical solution of nonlinear problems of fluid mech. by least squares. Comp.Meth. in Appl. Mech .. 17/18 March 1979.
[57] R. Brun: Transport et relaxation dans les ecoulements gazeux. Masson 1986.
[58] L. Cafarelli, R. Kohn, L. Nirenberg: On the regularity of the solutions of theNavier-Stokes equations. Comm. Pure and Appl. Math. , 35 , 771-831, 1982.
[59] J. Cahouet, J.P. Chabart: Some fast 3D finite element solvers for thegeneralized Stokes problem. Int. J. Numer. Methods. in Fluids . 8, 869-895.1988.
[60] Cahouet, Chabart: Some fast 3-D Finite Element solvers for GeneralizedStokes problem. Rapport EDF HE/41/87.03, 1987
[61] T. Chacon: Oscillations due to the transport of micro-structures. SIAM J.Appl. Math. 48 5 p1128-1146. 1988.
[62] T. Chacon, O. Pironneau: On the mathematical foundations of the k � ✏model. InVistas in applied mathematics . A.V. Balakrishnan ed. OptimizationSoftware inc. Springer, 1986.
[63] G. Chavent, G. Ja↵re: Mathematical methods and finite elements for reser-voir simulations . North-Holland, 1986.
[64] A.J. Chorin: Numerical study of slightly viscous flow. J. Fluid Mech . 57785-796. 1973.
[65] A. J. Chorin: A numerical Method for Solving Incompressible viscous flowproblems. J. Comp. Physics . 2, 12-26, 1967.
[66] I. Christie, D.F. Gri�ths, A.R. Mitchell, O.C. Zienkiewicz: Finite elementmethods for second order di↵erential equations with significant first derivatives.Int. J. Numer. Meth. Eng , 10 , 1389-1396, 1976.
120
[67] Ph. Ciarlet: The Finite Element Method. for Elliptic problems. North-Holland, 1978.
[68] P.G. Ciarlet Elasticite tridimensionnelle. Masson Ed., 1986.
[69] B. Cockburn: The quasi-monotone schemes for scalar conservation laws.IMA preprint 277 , University of Minnesota, Oct. 1986.
[70] G. Comte-Bellot: Ecoulement turbulent entre deux parois planes. Publica-tions scientifiques et techniques du ministere de l’air. 1980.
[71] P. Constantin, C. Fioas, O. Manley, R. Temam: Determining modes andfractal dimension of turbulent flows.J. Fluid Mech . 150 , 427-440, 1985.
[72] M. Crouzeix - P.A. Raviart: Conforming and non-conforming finite elementelement methods for the stationary Stokes eq. RAIRO . R3. 33-76, 1973 .
[73] C. Cuvelier, A. Segal, A.A. van Steenhoven: Finite Element Methods andNavier-Stokes equations. Mathematics and its applications series, D. ReidelPublishing Co. 1986.
[74] R. Dautray, J.L. Lions: Analyses mathematiques et calculs numeriques .Masson, 1987.
[75] J.W. Deardor↵: A numerical study of 3-d turbulent channel flow at largeReynolds numbers. J. Fluid Mech. 41 , 2, 453-480, 1970.
[76] A. Dervieux: Steady Euler simulations using unstructured meshes . VonKarman Lecture notes series 1884-04. Mars 1985.
[77] J.A. Desideri, A. Dervieux: Compressible Fluid Dynamics; compressible flowsolvers using unstructured grids . Von Karman lecture notes series. March 1988.
[78] Q. Dinh, R. Glowinski, J. Periaux: Domain decomposition for ellipticproblems . In Finite element in fluids . 5 R. Gallager, J. Oden, O. Zienkiewicz,T. Kawai, M. Kawahara eds. p45-106. Wiley. 1984
[79] J.M. Dominguez, A. Bendali, S. Gallic: A variational approach to the vectorpotential of the Stokes problem . J. Math. Anal. and Appl. 107,2, 537-560,1987.
[80] J. Donea: A Taylor-Galerkin method for convective transport problems. J.Numer. Meth. Eng, 20 , 101-120, 1984. (See also J. Donea, L. Quartapelle, V.Selmin: An analysis of time discretization in the Finite Element Solution ofHyperbolic problems. J. Comp. Physics, 70 2 1987.
121
[81] J. Douglas, T.F. Russell: Numerical methods for convection dominateddi↵usion problems based on combining the method of characteristics with finiteelement methods or finite di↵erence method. SIAM J. Numer Anal. 19, 5 871-885, 1982.
[82] J. M. Dupuy: Calculs d’ecoulements potentiels transsoniques; rapport in-terne Ecole Polytechnique 1986.
[83] A. Ecer, H. Akai: Computation of steady Euler equations using finite elementmethod. AIAA . 21 Nx3, p343-350. 1983.
[84] F. El-Dabaghi, J. Periaux, O. Pironneau, G. Poirier: 2d/3d finite elementsolution of the steady Euler equations. Int. J. Numer. Meth. in Fluid . 7 ,p1191-1209. 1987.
[85] F. El-Dabaghi, O. Pironneau: Stream vectors in 3D aerodynamics. Numer.Math. , 48 363-394, 1986.
[86] F. El-Dabaghi: Steady incompressible and compressible solution of Navier-Stokes equations by rotational correction. in Numerical met. for fluid dynamics.K.W.Morton and M.J. Baines. Clarendon press.p273-281. 1988.
[87] J. Essers: Computational methods for turbulent transonic and viscous flows.Springer, 1983.
[88] J. Feder: Fractals . Plenum Press. 1988.
[89] M.Festauer, J.Mandel, J.Necas: Entropy regularization of the Transonic Po-tential flow problem. Commentationes Mathematicae Universitiatis Carolinae.Prague, 1984. See also C. Necas: transonic flow Masson Masson 1988.
[90] F. Fezoui: Resolution des equations d’Euler par un schema de Van Leer enelements finis. Rapport Inria 358, 1985.
[91] C. Foias, R. Temam: Determination of the solution of the Navier-Stokesequations by a set of nodal values. Math. Comput ., 43 , 167, 117-133, 1984.
[92] M. Fortin: Calcul numerique des ecoulements par la methode des elements.Ph.D. Thesis , Universite Paris 6, 1976.
[93] M. Fortin, R. Glowinski: Augmented lagrangian methods . North-Holland1983.
[94] M. Fortin, A. Soulaimani: Finite element approximation of compressibleviscous flows. Proc. computational methods in flow analysis, vol 2, H. Niki andM. Kawahara ed. Okayama University Science press . (1988)
122
[95] M. Fortin, F. Thomasset: Mixed finite element methods for incompressibleflow problems. J. Comp. Physics . 31 113-145, 1979.
[96] U. Frisch, B. Hasslacher, Y. Pomeau: Lattice-gas automata for the Navier-Stokes equation.Physical Review letters . 56 , 14, 1505-1508, 1986.
[97] G.P. Galbi: Weighted residual energy methods in fluid dynamics . Springer1985.
[98] D. Gelder: Solution of the compressible flow equations. Int. J. Numer. Meth.Eng. (3) 35-43, 1987.
[99] W. Gentzsch: Vectorization of computer programs with application to CFD. Notes on numerical fluid mechanics. 8 Vieweg 1984.
[100] P. Geymonat, P. Leyland: Analysis of the linearized compressible Stokesproblem (to appear).
[101] J.M. Ghidaglia: On the fractal dimension of attractors for viscous incom-pressible fluid flows. SIAM J. Math. Anal. , 17, 1139-1157, 1986.
[102] K. Giannakoglou, P. Chavariopoulos, K. Papaliou: Numerical computationof 2D rotational inviscid transonic flows using the decomposition of vectorfields. 7th ISABE conf . Beijing. 1985.
[103] V. Girault: Finite Element Method for the Navier-Stokes equations withnon-standard boundary conditions in R3. Rapport du Laboratoire d’AnalyseNumerique de l’Universite Paris 6 n0 R87036, 1987.
[104] V.Girault, P.A. Raviart: Finite Element Approximations of the Navier-Stokes Eq . Lecture Notes in Math. Springer ,1979.
[105] V. Girault, P.A. Raviart: Finite Elements methods for Navier-Stokesequations . Springer series SCM vol 5, 1986.
[106] J. Glimm: Comm. Pure and Appl. Math . 18 . p697. 1965.
[107] R. Glowinski: Numerical methods for nonlinear variationnal problems.Springer Lectures Notes in Computationnal Physics, 1985.
[108] R. Glowinski: Le ✓ schema. Dans M.O. Bristeau, R. Glowinski, J. Periaux.Numerical methods for the Navier-Stokes equations. Comp. Phy. report 6 ,73-187, 1987.
[109] R. Glowinski, J. Periaux, O. Pironneau: An e�cient preconditioning schemefor iterative numerical solution of PDE.Mathematical Modelling,1979.
123
[110] R. Glowinski, O. Pironneau: Numerical methods for the first biharmonicproblem. SIAM Review 21 2 p167-212. 1979.
[111] J. Goussebaıle, A. Jacomy: Application a la thermo-hydrolique desmethodes d’eclatement d’operateur dans le cadre elements finis: traitementdu modele k � ✏. Rapport EDF-LNH HE/41/85.11, 1985.
[112] N. Goutal: Resolution des equations de Saint-Venant en regime transcri-tique par une methode d’elements finis. These , Universite Paris 6, 1987.
[113] P.M. Gresho-R.L. Sani: On pressure conditions for the incompressibleNavier-Stokes equations. in Finite Elements in Fluids 7 . R. Gallager et aled. Wiley 1988.
[114] P. Grisvard: Elliptic problems in non-smooth domains . Pitman, 1985.
[115] M. Gunzburger: Mathematical aspects of finite element methods for incom-pressible viscous flows. in Finite element, theory and application . D. Dwoyer,M Hussaini, R. Voigt eds. Springer ICASE/NASA LaRC series. 124-150 1988.
[116] L. Hackbush: The Multigrid Method: theory and applications. Springerseries SCM, 1986.
[117] M. Hafez, W. Habashi, P. Kotiuga: conservative calculation of non isen-tropic transonic flows. AIAA . 84 1929. 1983.
[118] L. Halpern: Approximations paraxiales et conditions absorbantes. Thesed’etat. Universite Paris 6. 1986.
[119] J. Happel, H. Brenner: Low Reynolds number hydrodynamics. Prentice Hall1965.
[120] F.H. Harlow, J.E. Welsh: The Marker and Cell method.Phys. Fluids 8,2182-2189, (1965).
[121] F. Hecht: A non-conforming P 1 basis with free divergence in R3. RAIROserie analyse numerique . 1983.
[122] J.C. Henrich , P.S. Huyakorn, O.C. Zienkiewicz, A.R. Mitchell: An upwindfinite element scheme for the two dimensional convective equation. Int. J. Num.Meth. Eng . 11 , 1831-1844, 1981.
[123] J. Heywood-R. Rannacher: Finite element approximation of the nonsta-tionnary Navier-Stokes equations (I) SIAM J. Numer. Anal . 19 p275. 1982.
[124] M. Holt: Numerical methods in fluid dynamics . Springer, 1984.
124
[125] P. Hood - G. Taylor: Navier-Stokes eq. using mixed interpolation. in Finiteelement in flow problem Oden ed. UAH Press,1974.
[126] K. Horiuti: Large eddy simulation of turbulent channel flow by one-equationmodeling. J. Phys. Soc. of Japan. 54 , 8, 2855-2865, 1985.
[127] T.J.R. Hughes:A simple finite element scheme for developing upwind finiteelements. Int. J. Num. Meth. Eng . 12 , 1359-1365, 1978.
[128] T. J.R. Hughes:The finite element method. Prentice Hall, 1987.
[129] T.J. R. Hughes, A. Brooks: A theoretical framework for Petrov-Galerkinmethods with discontinuous weighting functions: application to the streamlineupwind procedure. In Finite Elements in Fluids, R. Gallagher ed. Vol 4., Wiley,1983.
[130] T.J.R. Hughes, L.P. Franca, M. Mallet: A new finite element formulationfor computational fluid dynamics. Comp. Meth. in Appl. Mech. and Eng. 6397-112 (1987).
[131] T.J.R. Hughes, M. Mallet: A new finite element formulation for computa-tional fluid dynamics.Computer Meth. in Appl. Mech. and Eng. 54 ,341-355,1986.
[132] M. Hussaini, T. Zang: Spectral methods in fluid mechanics. Icase report86-25 . 1986.
[133] A. Jameson: Transonic flow calculations. In Numerical methods in fluidmech. H. Wirz, J. Smolderen eds. p1-87. McGraw-Hill. 1978.
[134] A. Jameson, T.J. Baker: Improvements to the Aircraft Euler Method.AIAA 25th aerospace meeting. Reno Jan 1987.
[135] A. Jameson, J. Baker, N. Weatherhill: Calculation of the inviscid transonicflow over a complete aircraft. AIAA paper 86-0103, 1986.
[136] C. Johnson: Numerical solution of PDE by the finite element method .Cambridge university press, 1987.
[137] C. Johnson: Streamline di↵usion methods for problems in fluid mechanics.Finite elements in fluids , Vol 6. R. Gallagher ed. Wiley, 1985.
[138] C. Johnson, U. Navert, J. Pitkaranta: Finite element methods for linearhyperbolic problems. Computer Meth. In Appl. Mech. Eng . 45 , 285-312, 1985.
125
[139] C. Johnson, J. Pitkaranta: An analysis of the discontinuous Galerkinmethod for a scalar hyperbolic equation. Math of Comp . 46 1-26, 1986.
[140] C. Johnson, A. Szepessy: On the convergence of streamline di↵usion finiteelement methods for hyperbolic conservation laws. in Numerical methods forcompressible flows , T.E. Tedzuyar ed. AMD-78 , 1987.
[141] Kato: Non stationary flows of viscous and ideal fluids in R3 .J. Func. Anal. 9 p296-305 ,1962.
[142] M. Kawahara, T. Miwa: Finite element analysis of wave motion. Int. J.Numer. Methods Eng . 20 1193-1210. 1986.
[143] M. Kawahara: On finite element in shallow water long wave flow analysis. inFinite element in nonlinear mechanics (J. Oden ed) North-Holland p261-287.1979.
[144] M. Kawahara: Periodic finite element method of unsteady flow. Int. J.Meth. in Eng. 11 7, p1093-1105. 1977.
[145] N. Kikuchi, T. Torigaki: Adaptive finite element methods in computedaided engineering. Danmarks Tekniske HØJSKOLE, Matematisk report 1988-09.
[146] S. Klainerman, A. Majda: Compressible and incompressible fluids. CommPure Appl. Math. 35, 629-651, 1982.
[147] R. Kohn, C. Morawetz: To appear. See also C. Morawetz: On a weaksolution for a transonic flow problem.Comm. pure and appl. math. 38 , 797-818,1985.
[148] S.N. Kruzkov: First order quasi-linear equations in several independentvariables. Math USSR Sbornik , 10 , 217-243, 1970.
[149] Y. A. Kuznetsov: Matrix computational processes in subspaces. in Comput-ing methods in applied sciences and engineering VI . R. Glowinski, J.L. Lionsed. North-Holland 1984.
[150] O. Ladyzhenskaya: The Mathematical Theory of viscous incompressibleflows . Gordon-Breach, 1963.
[151] O.A. Ladyzhenskaya, V.A. Solonnikov, N.N. Ouraltseva: Equationsparaboliques lineaires et quasi-lineaires . MIR 1967.
[152] H. Lamb: Hydrodynamics . Cambridge University Press, 1932.
126
[153] L. Landau- E. Lifschitz: Mecanique des Fluides . MIR Moscou, 1953.
[154] P. Lascaux, R. Theodor: Analyse numerique matricielle appliquee a l’artde l’ingenieur . Masson, 1986.
[155] B.E. Launder, D.B. Spalding: Mathematical models of turbulence . Aca-demic press, 1972.
[156] P.D. Lax: On the stability of di↵erence approximations to solutions ofhyperbolic equations with variable coe�cients.Comm. Pure Appl. Math . 9p267, 1961.
[157] P. Lax: Hyperbolic systems of conservation laws and the mathematicaltheory of shock waves , CMBS Regional conference series in applied math.11 , SIAM, 1972.
[158] J. Leray, C. Schauder: in H. Berestycki, These d’etat , Universite Paris 6,1982.
[159] A. Leroux: Sur les systemes hyperboliques. These d’etat. Paris 1979.
[160] P. Lesaint: Sur la resolution des systemes hyperboliques du premier ordrepar des methodes d’elements finis. These de doctorat d’etat, Univ. Paris 6 1975.
[161] P. Lesaint, P.A. Raviart: On a finite element method for solving the neutrontransport equation. In Mathematical aspect of finite elements in PDE . C. deBoor ed. Academic Press, 89-123, 1974.
[162] M. Lesieur: Turbulence in fluids . Martinus Nijho↵ publishers, 1987.
[163] J. Lighthill: Waves in fluids Cambridge University Press,1978.
[164] LINPACK, User’s guide . J.J. Dongara C.B. Moler, J.R. Bunch, G.W.Steward. SIAM Monograph, 1979.
[165] J.L. Lions: Quelques Methodes de Resolution des Problemes aux limitesNonlineaires . Dunod, 1968.
[166] J.L. Lions. Controle des systemes gouvernes par des equations aux deriveespartielles Dunod, 1969.
[167] J.L. Lions, E. Magenes:Problemes aux limites non homogenes et applications. Dunod 1968.
[168] J.L. Lions A. Lichnewsky: Super-ordinateurs: evolutions et tendances, lavie des sciences, comptes rendus de l’academie des sciences, serie generale,1,Octobre 1984.
127
[169] P.L. Lions: Solution viscosite d’EDP hyperbolique nonlineaire scalaire.Seminaire au College de France. Pitman 1989.
[170] R. Lohner: 3D grid generation by the advancing front method. In Laminarand turbulent flows . C. Taylor, W.G. Habashi, H. Hafez eds. Pinneridge press,1987.
[171] R. Lohner, K. Morgan, J. Peraire, O.C. Zienkiewicz: Finite elementmethods for high speed flows. AIAA paper . 85 1531.
[172] R. Lohner, K. Morgan, J. Peraire, M. Vahdati: Finite element flux-correctedtransport (FEM-FCT) for the Euler and Navier-Stokes equations. in FiniteElements in Fluids 7 . R. Gallager et al ed. Wiley 1988.
[173] Luo-Hong: Resolution des equations d’Euler en � ! en regimetranssonique. These Universite Paris 6. 1988.
[174] D. Luenberger: Optimization by vector space methods. Wiley 1969.
[175] R. MacCormack: The influence of CFD on experimental aerospace facilities;a fifteen years projection. Appendix C . p79-93. National academic press,Washington D.C. 1983.
[176] Y. Maday, A. Patera: Spectral element methods for the incompressibleNavier-Stokes equations. ASME state of the art survey in comp. mech. E. Noored. 1987.
[177] A. Majda: Compressible fluid flows and systems of conservation laws .Applied math sciences series, Springer Vol 53, 1984.
[178] M. Mallet: A finite element method for computational fluid dynamics.Doctoral thesis, University of Stanford, 1985.
[179] M. Mallet, J. Periaux, B. Stou✏et: On fast Euler and Navier-Stokes solvers.Proc. 7th GAMM conf. on Numerical Methods in Fluid Mechanics. Louvain.1987.
[180] A. Matsumura, T. Nishida: The initial value problem for the equationsof motion of viscous and heat conductive gases. J. Math. Kyoto Univ . 20 ,67-104, 1980.
[181] A. Matsumura, T. Nishida: Initial boundary value problems for the equa-tions of motion in general fluids, in Computer Methods in applied sciences andEngineering. R. Glowinski et al. eds; North-Holland, 1982.
128
[182] F.J. McGrath: Nonstationnary plane flows of viscous and ideal fluids.Arch.Rat. Mech. Anal . 27 p328-348, 1968.
[183] D. Mclaughin, G. Papanicolaou, O. Pironneau: Convection of microstruc-tures. Siam J. Numer. Anal . 45 ,5, p780-796. 1982.
[184] J.A. Meijerink- H.A. Van der Vorst: An iterative solution method for linearsystems of which the coe�cient matrix is a symmetric M-matrix. Math. ofComp . 31, 148-162 ,1977.
[185] P. Moin, J. Kim: Large eddy simulation of turbulent channel flow. J. Fluid.Mech . 118 , p341, 1982.
[186] K. Morgan, J. Periaux, F. Thomasset: Analysis of laminar flow over abackward facing step : Vieweg Vol 9: Notes on numerical fluid mechanics.1985.
[187] K. Morgan, J. Peraire, R. Lohner: Adaptive finite element flux correctedtransport techniques for CFD. in Finite element, theory and application. D.Dwoyer, M Hussaini, R. Voigt eds. Springer ICASE/NASA LaRC series. 165-174 1988.
[188] K.W. Morton: FEM for hyperbolic equations. In Finite Elements inphysics: computer physics reports Vol 6 No 1-6, R. Gruber ed. North Holland.Aout 1987.
[189] K.W. Morton, A. Priestley, E. Suli: Stability of the Lagrange-Galerkinmethod with non-exact integration. RAIRO-M2AN 22 (4) 1988, p625-654.
[190] F. Murat: l’injection du cone positif de H�1dans W�1,q est compacte pourtout q < 2.J. Math pures et Appl . 60 , 309-321, 1981.
[191] J.C. Nedelec: A new family of finite element in R3. Numer math . 50 ,57-82. 1986.
[192] J. Necas: Ecoulements de fluides; compacite par entropie. Masson. 1989.
[193] R.A. Nicolaides: Existence uniqueness and approximation for generalizedsaddle-point problems. SIAM J. Numer. Anal . 19 , 2, 349-357 (1981).
[194] J. Nitsche, A. Schatz: On local approximation properties of L2�projectionon spline-subspaces. Applicable analysis , 2 , 161-168 (1972)
[195] J.T. Oden, G. Carey: Finite elements; mathematical aspects . Prentice Hall,1983.
129
[196] J.T. Oden, T. Strouboulis, Ph. Devloo: Adaptive Finite element methodsfor high speed compressible flows. in Finite Elements in Fluids 7 . R. Gallageret al ed. Wiley 1988.
[197] S. Orszag: Statistical theory of turbulence.(1973). in Fluid dynamics . R.Balian-J.L.Peule ed. Gordon-Breach, 234-374, 1977.
[198] B. Palmerio, V. Billet, A. Dervieux, J. Periaux: Self adaptive mesh refine-ments and finite element methods for solving the Euler equations. Proceedingof the ICFD conf . Reading 1985.
[199] R.L. Panton: Incompressible flow . Wiley Interscience,1984.
[200] C. Pares: Un traitement faible par elements finis de la condition deglissement sur une paroi pour les equations de Navier-Stokes. Note CRAS .307 I p101-106. 1988.
[201] R. Peyret, T. Taylor: Computational methods for fluid flows. Springer seriesin computational physics, 1985.
[202] O. Pironneau: On the transport-di↵usion algorithm and its applications tothe Navier-Stokes equations. Numer. Math. 38 , 309-332, 1982.
[203] O. Pironneau: Conditions aux limites sur la pression pour les equations deNavier-Stokes. Note CRAS .303,i , p403-406. 1986.
[204] O. Pironneau, J. Rappaz: Numerical analysis for compressible isentropicviscous flows. (to appear in IMPACT, 1989)
[205] E. Polak: Computational methods in optimization. Academic Press, 1971.
[206] B. Ramaswami, N. Kikuchi: Adaptive finite element method in numericalfluid dynamics. in Computational methods in flow analysis . H. Niki, M.Kawahara ed. Okayama University press 52-62. 1988.
[207] G. Raugel: These de 3eme cycle, Universite de Rennes, 1978.
[208] G. Ritcher: An optimal order error estimate for discontinuous Galerkinmethods. Rutgers University, computer science report Fev. 1987.
[209] R. Ritchmeyer, K. Morton: Di↵erence methods for initial value problems.Wiley 1967.
[210] A. Rizzi, H.Bailey: Split space-marching finite volume method for chemi-cally reacting supersonic flow. AIAA Journal, 14 , 5, 621-628, 1976.
130
[211] A. Rizzi, H. Viviand eds: Numerical methods for the computation of inviscidtransonic flows with shock waves . 3 , Vieweg, 1981.
[212] W. Rodi: Turbulence models and their applications in hydrolics. Inst. Ass.for Hydrolics. State of the art paper. Delft. 1980.
[213] G. Roge: Approximation mixte et acceleration de la convergence pour lesequations de Navier-Stokes compressible en elements finis. These , UniversiteParis-6 (1989).
[214] E. Ronquist, A. Patera: Spectral element methods for the unsteady Navier-Stokes equation. Proc. 7th GAMM conference on Numerical methods in fluidmechanics . Louvain la neuve, 1987.
[215] L. Rosenhead ed.: Laminar Boundary layers. Oxford University Press.1963.
[216] Ph. Rostand: Kinetic boundary layers, numerical simulations. Inria report728,1987.
[217] Ph. Rostand, B. Stou✏et: TVD schemes to compute compressible viscousflows on unstructure meshes. Proc. 2nd Int Conf. on hyperbolic problems,Aachen (FRG) (To appear in Vieweg ) (1988).
[218] D. Ruelle: Les attracteurs etranges.La recherche , 108, p132 1980.
[219] D. Ruelle: Characteristic exponents for a viscous fluid subjected to timedependant forces.Com. Math. Phys . 93 , 285-300, 1984.
[220] I. Ryhming: Dynamique des fluides Presses Polytechniques Romandes,1985.
[221] B. Saramito: Existence de solutions stationnaires pour un probleme defluide compressible. DPH-PFC 1224 (1982).
[222] Y. Saad: Krylov subspace methods for solving unsymmetric linear systems.Math of Comp . 37 105-126, 1981.
[223] J.H. Saiac: On the numerical solution of the time-dependant Euler equa-tions for incompressible flow. Int. J. for Num. Meth. in Fluids . 5 , 637-656 ,1985.
[224] N. Satofuka: Unconditionally stable explicit method for solving the equa-tions of compressible viscous flows, Proc. 5th Conf. on numer. meth. in fluidmech . 7 , Vieweg. 1987.
131
[225] V. Schumann: subgrid scale model for finite di↵erence simulations ofturbulent flows in plane channel and annuli. J. Comp. Physics. 18 376-404,1975.
[226] L.R. Scott, M. Vogelius: Norm estimates for a maximal inverse of thedivergence operator in spaces of piecewise polynomials. RAIRO M2AN . 19111-143. 1985.
[227] V.P. Singh: Resolution des equations de Navier-Stokes en elements finis pardes methodes de decomposition d’operateurs et des schemas de Runge-Kuttastables. These de 3eme cycle , universite Pars 6, 1985.
[228] J.S. Smagorinsky: General circulation model of the atmosphere. Mon.Weather Rev . 91 99-164, 1963.
[229] A. Soulaimani, M. Fortin, Y. Ouellet, G. Dhatt, F. Bertrand: Simplecontinuous pressure element for 2D and 3D incompressible flows. Comp. Meth.in Appl. Mech. and Eng . 62 47-69 (1987).
[230] C. Speziale: On non-linear K-l and K � ✏ models of turbulence. J. FluidMech. 178 p459. 1987.
[231] C. Speziale:Turbulence modeling in non-inertial frames of references.ICASE report 88-18.
[232] R. Stenberg: Analysis of mixed finite element methods for the Stokesproblem; a unified approach. Math of Comp .42 9-23. 1984.
[233] B. Stou✏et: Implicite finite element methods for the Euler equations. inNumerical method for the Euler equations of Fluid dynamics. F. Angrand ed.SIAM series 1985.
[234] B.Stou✏et, J. Periaux, L. Fezoui, A. Dervieux: Numerical simulations of3D hypersonic Euler flows around space vehicles using adaptive finite elementsAIAA paper 8705660, 1987.
[235] G. Strang, G. Fix: An analysis of the finite element method . Prentice Hall1973.
[236] A.H. Stroudl: Approximate calculation of multiple integrals . Prentice-Hall1971.
[237] C. Sulem, P.L Sulem, C. Bardos, U. Frisch: Finite time analycity for the3D and 3D Kelvin-Helmholtz instability. Comm. Math. Phys. , 80 , 485-516,1981.
132
[238] E. Suli: Convergence and non-linear stability of the Lagrange-Galerkinmethod for the Navier-Stokes equations. Numer. Math. 53 p459-483, 1988.
[239] T.E. Tezduyar, Y.J. Park, H.A. Deans: Finite element procedures for timedependent convection-di↵usion-reaction systems. in Finite Elements in Fluids7 . R. Gallager et al ed. Wiley 1988.
[240] R. Temam:Theory and Numerical Analysis of the Navier-Stokes eq . North-Holland 1977.
[241] F. Thomasset: Implementation of Finite Element Methods for Navier-Stokes eq . Springer series in Comp. Physics 1981.
[242] N. Ukeguchi, H. Sataka, T. Adachi: On the numerical analysis of com-pressible flow problems by the modified ’flic’ method. Comput. fluids. 8, 251,1980.
[243] A. Valli: An existence theorem for compressible viscous flow. Boll Un. Mat.It. Anal. Funz. Appl . 18-C, 317-325, 1981.
[244] B. van Leer: Towards the ultimate conservation di↵erence scheme III. J.Comp. Physics . 23 , 263-275, 1977.
[245] R. Verfurth: Finite element approximation of incompressible Navier-Stokesequations with slip boundary conditions. Numer Math 50 ,697-721,1987.
[246] R. Verfurth: A preconditioned conjugate gradient residual algorithm forthe Stokes problem. in R. Braess, W. Hackbush, U. Trottenberg (eds) Devel-opments in multigrid methods , Wiley, 1985.
[247] P.L. Viollet: On the modelling of turbulent heat and mass transfers forcomputations of buoyancy a↵ected flows. Proc. Int. Conf. Num. Meth. forLaminar and Turbulent Flows . Venezia, 1981.
[248] A. Wambecq: Rational Runge-Kutta methods for solving systems of ODE,Computing , 20 , 333-342, 1978.
[249] L. Wigton, N. Yu, D. Young: GMRES acceleration of computational fluiddynamic codes. AIAA paper 85-1494, p67-74. 1985.
[250] P. Woodward, Ph. Colella: The numerical simulation of 2D fluid
flows with strong shocks. J. Comp. Physics ,54 , 115-173, 1984.
[251] S.F.Wornom M. M. Hafez: Implicit conservative schemes for the Eulerequations. AIAA Journal 24 ,2, 215-223, 1986.
133
[252] N. Yanenko: The method of fractional steps . Springer. 1971.
[253] O.C. Zienkiewicz:The finite element method in engineering science.McGraw-Hill 1977, Third edition.
[254] W. Zilj: Generalized potential flow theory and direct calculation of veloc-ities applied to the numerical solution of the Navier-Stokes equations. Int. J.Numer. Meth. in Fluid 8 , 5, 599-612. 1988.
[255] Arnold D. An Interior Penalty FEM with Discontinuous Elements. SIAMJ. Numer. Anal. 19, 742-760 (1982).
[256] Zhangxin Chen Finite Element Methods and Their Applications. Springerseries in Scientific computation (2005).
[257] Douglas J. and Dupont T. : Interior Penalty Procedures for Elliptic andParabolic Problems. Lecture notes in physics vol 58. p207-216. Springer (1976).
[258] Lesaint P. and P-A. Raviart: On a FEM for solving the Neutron TransportEquation. in Math Aspects of FEM for PDE. C. de Boor ed. Academic Press(1974).
[259] Delves L., Hall C. An implicit matching principle for global elementcalculations. J. Inst. Math. App. 23,223-234.(1979).
[260] Oden J. and I. Babuska and C. Baumann: A Discontinuous hp FiniteElement Method for Di↵usion Problems. J. Comp. Physics 146, 491-519 (1998).
[261] Riviere B. and M. Wheeler and V. Girault : Improved Energy Estimates forInterior Penalty, constrained and Discontinuous Galerkin Methods for EllipticSystems. Part I. Comput. Geosc. 3, 337-360 (1999).
[262] Wheeler M. : An Elliptic Collocation FEM with Interior Penalties. AIAMJ. Numer. Anal. 15, 152-161 (1978).