computational gas dynamics

8/10/2019 Computational Gas Dynamics

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D R

A F TComputational Gas Dynamics

January - April, 2011

Dr. S.V. Raghurama RaoAssociate Professor

CFD Centre

Department of Aerospace EngineeringIndian Institute of ScienceBangalore 560012, India

E-mails : (i) [email protected](ii) [email protected]


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D R

A F T

ii


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RAFT -- DRAFT -- DRAFT -- DRAFT -- DRAFT

Contents

1 Introduction 1

2 Compressible Fluid Flows, Their Governing Equations, Models and Ap-proximations 32.1 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Euler Equations, Burgers Equation and Linear Convection Equation . . . . 5

2.2.1 Euler equations in one dimension . . . . . . . . . . . . . . . . . . . 52.2.2 Euler equations in primitive variable form . . . . . . . . . . . . . . 62.2.3 Isentropic Euler Equations . . . . . . . . . . . . . . . . . . . . . . . 82.2.4 Isentropic Euler equations in characteristic form . . . . . . . . . . . 102.2.5 Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.6 Linear Convection Equation . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Linear Convection Equation, Characteristics and Hyperbolicity . . . . . . . 162.4 Burgers Equation, Shock Waves and Expansion Waves . . . . . . . . . . . 202.5 Shock Waves in Supersonic Flows . . . . . . . . . . . . . . . . . . . . . . . 212.6 Mathematical Classication of PDEs . . . . . . . . . . . . . . . . . . . . . 22

2.6.1 First Order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6.2 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.6.3 Second Order PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6.4 Physical Signicance of the Classication . . . . . . . . . . . . . . . 29

2.7 Euler equations and Hyperbolicity . . . . . . . . . . . . . . . . . . . . . . . 332.8 Kinetic Theory, Boltzmann Equation and its Moments as Macroscopic

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.8.1 B-G-K Model for the collision term . . . . . . . . . . . . . . . . . . 422.8.2 Splitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.9 Relaxation Systems for Non-linear Conservation Laws . . . . . . . . . . . . 442.9.1 Chapman-Enskog type expansion for the Relaxation System . . . . 462.9.2 Diagonal form of the Relaxation System . . . . . . . . . . . . . . . 482.9.3 Diagonal form as a Discrete Kinetic System . . . . . . . . . . . . . 502.9.4 Multi-dimensional Relaxation Systems . . . . . . . . . . . . . . . . 52

2.10 A Note on Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . 55

iii


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iv CONTENTS

3 Analysis of Convection Equations 57

3.1 Linear Convection Equation and Method of Characteristics . . . . . . . . . 573.2 Linear Convection Equation and Travelling Waves . . . . . . . . . . . . . . 613.3 Linear Convection Equation with a Variable Coefficient as Convection Speed 623.4 Non-linear Convection Equation . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Gradient Catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.6 Piece-wise Smooth Solutions and Discontinuities for Non-linear Convection

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 753.7 Weak Solutions and Integral Forms of Conservation Laws . . . . . . . . . . 773.8 Shock Wave Solution of a Non-linear Convection Equation and the Rankine-

Hugoniot (Jump) condition . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.9 Expansion Waves with Non-linear Convection Equations . . . . . . . . . . 863.10 Non-unique Solutions of Non-linear Convection Equations . . . . . . . . . . 953.11 Entropy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

3.11.1 Entropy Condition Version - I : . . . . . . . . . . . . . . . . . . . . 983.11.2 Entropy Condition Version - II : . . . . . . . . . . . . . . . . . . . . 993.11.3 Entropy Condition Version - III : . . . . . . . . . . . . . . . . . . . 993.11.4 Entropy Condition Version - IV : . . . . . . . . . . . . . . . . . . . 100

4 Analysis of Numerical Methods 1034.1 Basics of Finite Difference and Finite Volume Methods . . . . . . . . . . . 103

4.1.1 Upwind Method in Finite Difference Form . . . . . . . . . . . . . . 1034.1.2 Upwind Method in Finite Volume Form . . . . . . . . . . . . . . . 1064.2 Modied Partial Differential Equations . . . . . . . . . . . . . . . . . . . . 1104.3 Consistency of Numerical Methods . . . . . . . . . . . . . . . . . . . . . . 1144.4 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.4.1 Fourier series in complex waveform . . . . . . . . . . . . . . . . . . 1154.4.2 Stability analysis of Numerical Methods . . . . . . . . . . . . . . . 116

5 Central Discretization Methods for Scalar and Vector Conservation Laws1235.1 A Brief History of Numerical Methods for Hyperbolic Conservation Laws . 1235.2 Lax-Friedrichs Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Lax-Wendroff Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1265.4 Two-Step Lax-Wendroff Method and MacCormack Method . . . . . . . . . 130

6 Upwind Methods for Scalar Conservation Laws 1336.1 Flux Splitting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.2 Approximate Riemann Solver of Roe . . . . . . . . . . . . . . . . . . . . . 136

6.2.1 Entropy Fix for Roes Scheme . . . . . . . . . . . . . . . . . . . . . 1396.3 Kinetic Flux Splitting Method . . . . . . . . . . . . . . . . . . . . . . . . . 1426.4 Relaxation Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147


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CONTENTS v

6.4.1 Relaxation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.4.2 Discrete Kinetic Scheme . . . . . . . . . . . . . . . . . . . . . . . . 1496.4.3 A Low Dissipation Relaxation Scheme . . . . . . . . . . . . . . . . 151

7 Upwind Methods for Vector Conservation Laws 1577.1 Some features of Euler equations . . . . . . . . . . . . . . . . . . . . . . . 157

7.1.1 Homogeneity Property . . . . . . . . . . . . . . . . . . . . . . . . . 1577.1.2 Linear Convection Equation: . . . . . . . . . . . . . . . . . . . . . . 1587.1.3 Burgers equation: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1597.1.4 Euler Equations: . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.2 Flux Vector Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1617.2.1 Flux Splitting for Linear Convection Equation . . . . . . . . . . . . 1617.2.2 Flux Vector Splitting for Euler Equations: . . . . . . . . . . . . . . 162

7.3 Approximate Riemann Solver or Roe . . . . . . . . . . . . . . . . . . . . . 1697.3.1 Projection of the solution onto the eigenvector space . . . . . . . . 1697.3.2 Roes Approximate Riemann Solver . . . . . . . . . . . . . . . . . . 1727.3.3 Roe averages with the use of a parameter vector . . . . . . . . . . . 1817.3.4 Properties of the Roe Linearization . . . . . . . . . . . . . . . . . . 1857.3.5 Wave strengths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1887.3.6 Solution of the Approximate Riemann Problem . . . . . . . . . . . 193

7.4 Kinetic Flux Vector Splitting Method for Euler Equations . . . . . . . . . 1977.5 Peculiar Velocity based Upwind Method for Euler Equations . . . . . . . . 202


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vi CONTENTS


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Chapter 1

Introduction

Computational Fluid Dynamics (CFD) is the science and art of simulating uid ows oncomputers. Traditionally, experimental and theoretical uid dynamics were the two di-mensions of the subject of Fluid Dynamics. CFD has emerged as a third dimension in thelast three decades [1]. The rapid growth of CFD as a design tool in several branches of en-gineering, including Aerospace, Mechanical, Civil and Chemical Engineering, is due to theavailability of fast computing power in the last few decades, along with the developmentof several intelligent algorithms for solving the governing equations of Fluid Dynamics.The history of the development of numerical algorithms for solving compressible uidows is an excellent example of the above process. In this course, several important andinteresting algorithms developed in the past three decades for solving the equations of compressible uid ows are presented in detail.

In the next chapter (2 nd chapter), the basic governing equations of compressible uidows are described briey, along with some simplications which show the essential na-ture of the convection process. The basic convection process is presented in terms of simpler scalar equations to enhance the understanding of the convection terms. It is thepresence of the convection terms in the governing equations that makes the task of de-veloping algorithms for uid ows difficult and challenging, due to the non-linearity. Thebasic convection equations are presented in both linear and non-linear forms, which willform the basic tools for developing and testing the algorithms presented in the later chap-ters. The hyperbolic nature of the convection equations is presented in detail, along withits important implications. Some of the numerical methods presented in later chaptersdepend upon deriving the equations of compressible ows from Kinetic Theory of gasesand as Relaxation Approximations. Therefore, these systems are also presented in thischapter, as a part of the governing equations for compressible ows, their models andapproximations.

In the third chapter, the basic tools required for analyzing the numerical methods -consistency and stability of numerical methods, modied partial differential equations,numerical dissipation and dispersion, order of accuracy of discrete approximations - are

1


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2 Introduction

briey presented. These tools help the student in understanding the algorithms presented

in later chapters better.The fourth chapter presents the central discretization methods for the hyperbolic equa-tions, which were the earliest to be introduced historically. The fth chapter presents theupwind discretization methods which became more popular than the central discretizationmethods in the last two decades. The four major categories of upwind methods, namely,Riemann Solvers, Flux Splitting Methods, Kinetic Schemes and Relaxation Schemes, arepresented for the scalar conservation equations in this chapter. More emphasis is given tothe alternative formulations of recent interest, namely, the Kinetic Schemes and Relax-ation Schemes. The corresponding numerical methods for vector conservation equationsare presented in the next chapter.


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Chapter 2

Compressible Fluid Flows, Their

Governing Equations, Models andApproximations

2.1 Navier-Stokes EquationsThe governing equations of compressible uid ows are the well-known Navier-Stokesequations. They describe the conservation of mass, momentum and energy of a owinguid. In three dimensions, we can write the Navier-Stokes equations in the vector formas

U t

+ G1

x +

G2y

+ G3

z =

G1,V x

+ G2,V

y +

G3,V z

(2.1)

where U is the vector of conserved variables, also known as the vector of state variables,dened by

U =

u1u2u3

E

(2.2)

G1, G2 and G3 are the inviscid ux vectors, given by

G1 =

u1 p + u21

u1u2u1u3

pu1 + u1E

; G2 =

u2u2u1

p + u22u2u3

pu2 + u2E

; G2 =

u3u3u1u3u2

p + u23 pu2 + u3E

(2.3)

G1,V , G2,V and G3,V are the viscous ux vectors, given by

3


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4 Governing Equations and Approximations

G1,V =

0 xx xy xz

u1 xx + u2 xy + u3 xz q 1; G2,V =

0 yx yy yz

u1 yx + u2 yy + u3 yz q 2

G3,V =

0 zx zy

zzu1 zx + u2 zy + u3 zz q 3(2.4)

In the above equations, is the density of the uid, u1, u2 and u3 are the components of the velocity in x,y and z directions, p is the pressure and E is the total energy (sum of internal and kinetic energies) given by

E = e + 12

u21 + u22 + u

23 (2.5)

The internal energy is dened bye =

p

( 1) (2.6)

where is the ratio of specic heats ( = C P C V ). The pressure, temperature and the densityare related by the equation of state as p = RT where R is the gas constant per unit mass,obtained by dividing the universal gas constant by molecular weight (for air at standardconditions, R = 217 J/ (Kg K )) and T is the temperature. Here, ij (i=1,2,3, j =1,2,3)represents the shear stress tensor and q i (i=1,2,3) represents the heat ux vector, theexpressions for which are as follows.

xx = 2u 1x

+ bulku 1x

+ u2y

+ u3z

(2.7)

yy = 2u2

y + bulk u

1

x + u2

y + u3

z (2.8)

zz = 2u 3z

+ bulku 1x

+ u2y

+ u3z

(2.9)

xy = yx = u 1y

+ u2x

(2.10)

xz = zx = u 1z

+ u3x

(2.11)


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Euler equations and simplications 5

yz = zy = u 3

y +

u2

z (2.12)

q 1 = kT x

, q 2 = kT y

, q 3 = kT z

(2.13)

where is the coefficient of viscosity, bulk is the bulk viscosity coefficient dened bybulk =

23

and k is the thermal conductivity.

2.2 Euler Equations, Burgers Equation and LinearConvection Equation

2.2.1 Euler equations in one dimensionA simplication which is often used is the inviscid approximation in which the viscous andheat conduction effects are neglected. This approximation is valid in large parts of theuid ows around bodies, except close to the solid surfaces where boundary layer effectsare important. The equations of inviscid compressible ows, called Euler equations , areobtained by neglecting the right hand side of the Navier-Stokes equations (2.1). In thiscourse, we shall often use the Euler equations and their further simplications.

Consider the 1-D Navier Stokes equations given by

U t

+ Gx

= GV x

(2.14)

where

U =

uE

, G =u

p + u2 pu + uE

and GV =0

u q (2.15)

Here, is the 1-D component of the stress tensor and q is the corresponding componentof the heat ux vector. They are dened for this 1-D case by

= 43

ux

and q = kT x

(2.16)

where is the viscosity of the uid and k is the thermal conductivity. Let us make therst simplifying assumption of neglecting the viscosity and heat conduction. Then, weobtain the 1-D Euler equations as

U t

+ Gx

= 0 (2.17)


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2.2.2 Euler equations in primitive variable formLet us now expand the conservative form of Euler equations to obtain the primitivevariable form. The mass conservation equation

t

+ (u)

x = 0 (2.18)

givest

+ ux

+ ux

= 0 (2.19)

The momentum conservation equation

(u)t

+ ( p + u2)

x = 0 (2.20)

gives

ut

+ ut

+ px

+ (u2)

x + u2

x

= 0

or

ut

+ ut

+ px

+ 2uux

+ u2x

= 0

or

ut + u2

x + u

ux +

ut + u

ux +

px = 0

or

ut

+ ux

+ ux

+ ut

+ uux

+ 1

px

= 0 (2.21)

Using (2.19), we obtain from the above equation

ut

+ uux

+ 1

px

= 0 (2.22)

The energy conservation equation

(E )t

+ ( pu + uE )

x = 0 (2.23)

gives

E t

+ E t

+ ( pu)

x + u

E x

+ E (u)

x = 0

or

E t

+ uE x

+ ( pu)

x + E

t

+ (u)

x = 0


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which, after using the mass conservation equation (2.18), becomes

E t

+ uE x

+ u

px

+ p

ux

= 0 (2.24)

SinceE =

p ( 1)

+ 12

u2

we obtainE t

= 1 1

t

p1 + 12

t

u2

= 1

1

p2

t

+ 1

1

1

pt

+ uut

(2.25)

andE x

= 1 1

p2

x

+ 1 1

1

px

+ uux

(2.26)

Therefore, the energy equation becomes

1

1 p2

t

1 1

1

pt

+ uut

1 1

pu2

x

1 1

u

px

+ u2ux

+ u

px

+ p

ux

= 0

or

p

t+

p

t+(

1) u

u

t pu

x+ u

p

x+(

1) u2

u

x+(

1) u

p

x+(

1) p

u

x = 0

or

p

t

pu

x

+pt

+ upx

+( 1) uut

+( 1) u2ux

+( 1) upx

+( 1) pux

= 0

or

p

t

+ ux

+ pt

+ upx

+ ( 1) uut

+ u2ux

+ upx

+ pux

= 0

Using (2.19), we obtain

p

ux

+ pt

+ upx

+ pux

+ ( 1) uut

+ uux

+ 1

px

+ ( 1) pux

= 0

which, after using (2.22), gets simplied to

pt

+ upx

+ pux

+ ( 1) pux

= 0

orpt

+ upx

+ pux

[1 + 1] = 0


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orpt + u

px + p

ux = 0 (2.27)

Thus, the 1-D Euler equations in primitive variable form (equations (2.19), (2.22) and(2.27)) can be written as

V t

+ AV x

= 0 (2.28)

where

V =u

p

and A =

u 0

0 u 1

0 p u

(2.29)

The eigenvalues of A are determined by

|A I | = 0or

u 00 u 10 p u

= 0

or(u ) (u )

2

p

= 0

or(u ) (u )

2

a2 = 0 as a2 = p

Thus, the eigenvalues of A are given by

1 = u a ; 2 = u ; 3 = u + a (2.30)

2.2.3 Isentropic Euler EquationsLet us consider the energy equation in primitive variables, given by

pt

+ upx

+ pux

= 0 (2.31)

Using the denition of the sound speed, a, as

a2 = p


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we obtainpt + u

px + a2

ux = 0 (2.32)

The entropy S is dened by

S = C V ln p

+ constant (2.33)

Therefore, the derivatives of S are given by

S t

= C V a2

pt

+ 1 p

pt

(2.34)

and S x

= C V a2

px

+ 1 p

px

(2.35)

from which we obtainpt

= pC V

S t

+ a2t

(2.36)

andpx

= pC V

S x

+ a2x

(2.37)

Using the above two equations, the energy equation can be written as

pC V

S t

+ a2 t

+ upC V

S x

+ ua2 x

+ a2 ux

= 0

orS t

+ C V p

a2t

+ C V p

pC V

uS x

+ C V p

a2 ux

+ ux

= 0

orS t

+ uS x

+ C V p

a2t

+ ux

+ ux

= 0

which becomesS

t + u

S

x = 0 (2.38)

when the continuity equation (2.22) is used. Thus, for isentropic ow, the Euler equationsare obtained as

t

+ ux

+ ux

= 0

ut

+ uux

+ 1

px

= 0

S t

+ uS x

= 0

(2.39)


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2.2.4 Isentropic Euler equations in characteristic formConsider the isentropic Euler equations in 1-D, given by

t

+ ux

+ ux

= 0

ut

+ uux

+ 1

px

= 0

S t

+ uS x

= 0

(2.40)

Let us change the primitive variables from ( , u, S ) to ( p, u, S ). Since the sound speed

a is dened bya2 =

p

(2.41)

we havep = a2

Thereforept

= a2t

(2.42)

andpx = a

2 x (2.43)

The continuity equation, given by

t

+ ux

+ ux

= 0

then becomes1a2

pt

+ u 1a2

px

+ ux

= 0

orpt + u

px + a2

ux = 0 (2.44)

The 1-D isentropic Euler equations then can be written as

pt

+ upx

+ a2ux

= 0

ut

+ uux

+ 1

px

= 0

S t

+ uS x

= 0

(2.45)


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Note that the third equation in the above system of equations is already in characteristic

form (being similar to a rst order wave equation). For this equationdxdt

= u (2.46)

is the equation of the characteristic curve and along this characteristic curve

dS dt

= 0 (2.47)

which means that the entropy is constant along these characteristic paths. Let us tryto write the rst two equations of (2.45) in characteristic form. Multiplying the second

equation of (2.45) by a and adding it to the rst equation, we obtainpt

+ upx

+ a2ux

+ aut

+ uux

+ 1

px

= 0

or

pt + ( u + a) px + a ut + ( u + a) ux = 0 (2.48)Similarly, multiplying the second equation of (2.45) by a and subtracting it from therst equation, we obtain

pt + u px + a2 ux a ut + uux + 1 px = 0

or

pt + ( u a) px a ut + ( u a) ux = 0 (2.49)Dividing both the above equations by a and rearranging, we get

ut

+ 1a

pt

+ ( u + a)ux

+ 1a

px

= 0 (2.50)

and ut 1a pt + ( u a) ux 1a px = 0 (2.51)Let us now dene w1 and w2 with

dw1 = du + 1a

dp (2.52)

anddw2 = du

1a

dp (2.53)


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which give

w1t

= ut

+ 1a

pt

w1x

= ut

+ 1a

px

w2t

= ut

1a

pt

w2x

= ux

1a

px

(2.54)

We then obtainw1t + ( u + a)

w1x = 0 (2.55)

andw2t

+ ( u a) w2x

= 0 (2.56)

which are in characteristic form, with the constants along the characteristic curves beingdened by

w1 = constant along dxdt

= ( u + a) (2.57)

andw2 = constant along

dx

dt = ( u

a) (2.58)

The 1-D isentropic Euler equations then become

w1t

+ ( u + a) w1x

= 0 with dw1 = du + 1a

dp

w2t

+ ( u a) w2x

= 0 with dw2 = du 1a

dp

w3t

+ uw3x

= 0 with w3 = S

(2.59)

Let us obtain the expressions of w1 and w2. Since

dw1 = du + 1adp

we get

w1 = u + 1a dpUsing the denition of the sound speed

a2 = dpd


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we can write

dp = a2d

Thereforew1 = u + 1a a2d = u + ad

From the denition of the entropy

S = C V ln p

+ k where k is a constant

we get

ln p =

S

k

C V or

p

= expS k

C V = e

SC V k where k = k

C V = a constant

Therefore p = e

SC V k where k = ek = a constant

Thus p = k e

SC V (2.60)

For isentropic ow, since S is a constant. Therefore, we get

p = c where c = a constant (2.61)

Thena2 =

dpd

= dd

(c ) = c( a )

ora = ( c )

12

( 1)2 (2.62)

Thusw1 = u +

a

d

= u + 1 (c )

1

2 ( 1)

2 d= u + ( c )

12

( 1)2 1d= u + ( c )

12

( 3)2 d

= u + ( c )12

( 1)21

12= u +

2( 1)

(c )12

( 1)2

= u + 2

( 1)a


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where we have used the equation (2.62). Therefore, we obtain

w1 = u + 2a( 1)

(2.63)

Similarly, we getw2 = u

2a( 1)

(2.64)

Therefore, the characteristic form of the isentropic Euler equations are given by

w1t

+ ( u + a) w1x

= 0 where w1 = u + 2a

(

1)

w2t

+ ( u a) w2x = 0 where w2 = u 2a( 1)w3t

+ uw3x

= 0 where w3 = S

(2.65)

2.2.5 Burgers equationOf the three equations in the system of 1-D isentropic Euler equations in characteristicform (equations (2.65), the rst two equations can be simplied further, by assuming theuid to be a mono-atomic gas.

= C P C V

= D + 2 + 2D + 2

for a poly-atomic gas (2.66)

and =

C P C V

= D + 2

D for a mono-atomic gas (2.67)

Thus, for air ow in 3-D, = 3 + 2 + 2

3 + 2 =

75

= 1.4. For a mono-atomic gas ow in 1-D,we obtain

= 1 + 2

1 = 3 (2.68)

Therefore, we getw1 = u +

2a 1

= u + 2a3 1

= u + a (2.69)

andw2 = u

2a 1

= u 2a3 1

= u a (2.70)The rst two equations of (2.65) then become

w1t

+ w1w1x

= 0 (2.71)


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and

w2t + w2 w2

x = 0 (2.72)

These two equations are of the form

ut

+ uux

= 0 (2.73)

This equation is known as the inviscid Burgers equation . Let us now put the aboveequation in conservative form as

ut +

g (u)x = 0 where g (u) =

u2

2 (2.74)

So, the ux g (u) is a quadratic function of the conservative variable u. That is, g (u)is not a linear function of u and hence it is a non-linear equation. The non-linearity of the convection terms is one of the fundamental difficulties in dealing with NavierStokesequations.

2.2.6 Linear Convection Equation

For the sake of simplicity, let us linearize the ux g(u) in the above equation as

g (u) = cu where c is a constant (2.75)

We are doing this linearization only to study and understand the basic convection terms.When we try to solve the Euler or NavierStokes equations, we will use only the non-linear equations. With the above assumption of a linear ux, the inviscid Burgers equationbecomes

ut

+ cux

= 0 (2.76)

This is called as the linear convection equation . So, the linear convection equation rep-resents the basic convection terms in the NavierStokes equations. The researchers inCFD use the numerical solution of the linear convection equation as the basic buildingblock for developing numerical methods for Euler or NavierStokes equations. The Ki-netic Schemes and the Relaxation Schemes , the two alternative numerical methodologieswhich will be presented in this course, also exploit this strategy, but in a different manner,as the Boltzmann equation and the Discrete Boltzmann equation, without the collisionterm, are just linear convection equations. Note also that the third of the 1-D isentropicEuler system (2.65) is just a linear convection equation. Thus, the Euler equations consistof two inviscid Burgers equations (non-linear PDEs) and one linear convection equation.


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16 Governing equations and approximations

2.3 Linear Convection Equation, Characteristics andHyperbolicity

To understand the nature of the linear convection equation better, let us rst nd out itssolution. The linear convection equation

ut

+ cux

= 0 (2.77)

is a rst order wave equation, also called as advection equation . It is a rst order hyperbolicpartial differential equation. Hyperbolic partial differential equations are characterised byinformation propagation along certain preferred directions. To understand this better, letus derive the exact solution of (2.77), given the initial condition

u(x, t = 0) = u0(x) (2.78)

Let us now use the method of characteristics to nd the value of the solution, u(x, t ), ata time t > 0. The method of characteristics uses special curves in the x t plane alongwhich the partial differential equation (PDE) becomes an ordinary differential equation(ODE). Consider a curve in the x t plane, given by (x(t), t ). The rate of change of u

t

(x(t),t)

(0,0) x

Figure 2.1: 3-Point Stencil

along this curve is given by ddt

u (x (t) , t ). Using chain rule, we can write

ddt

u (x (t) , t ) = x

u (x (t)) dxdt

+ t

u (x (t) , t ) (2.79)

which can be written simply as

dudt

= ut

+ dxdt

ux

(2.80)


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Linear convection equation and characteristics 17

The right hand side of (3.4) looks similar to the left hand side of the linear convection

equation (2.77), i.e., ut + cux . Therefore, if we choose

c = dxdt

(2.81)

then (2.77) becomes dudt

= 0, which is and ODE! The curve (x (t) , t ), therefore, should bedened by

dxdt

= c with x (t = 0) = xo (2.82)

The solution of (3.6) is given by x = ct + k (2.83)

where k is a constant. Using the initial condition x (t = 0) = x0, we get

x0 = k (2.84)

Thereforex = ct + x0 (2.85)

The curve x = ct + x0 is called the characteristic curve of the equation ut

+ cux

= 0

(2.77), or simply as the characteristic . Along the characteristic, the PDE, ut

+ cux

= 0,

becomes and ODE, dudt

= 0. The solution of this ODE is u = constant . Therefore, alongthe characteristic, the solution remains a constant. Thus, if we know the solution at thefoot of the characteristic (at x0), which is the initial condition, we can get the solutionanywhere on the characteristic, that is, u (x, t ). Using (3.9), we can write

x0 = x ct (2.86)Therefore

u (x, t ) = u0 (x0) = u0 (x ct) (2.87)We can write (3.9) asct = x x0 or t =

1c

x x0

c or t =

1c

x k (2.88)where k is a constant. Therefore, we can sketch the characteristics in the x t plane.They are parallel straight lines with slope

1c

. For non-linear PDES, characteristics neednot be parallel straight lines.


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s l o p

e = 1 /

c

t

x


Let us now derive the solution (3.11) in a mathematical way. For the PDE which is

the linear convection equation given by ut

+ cux

= 0, let us choose dxdt

= c. Its solutionis x = ct + x0 or x0 = x ct. This suggests a transformation of coordinates from ( x, t ) to(s, ) where

s = x c and = t (2.89)The inverse transformation is given by

x = s + c and t = (2.90)

Therefore, the transformation is from u (x, t ) to u (s, ). Since the function is the samein different coordinate systems

u (x, t ) = u (s, ) (2.91)

We can now write

dud

= ut

t

+ ux

x

= ut

t

+ ux

x

(since u = u) (2.92)

From (3.14), we have

t = 1 and x = c (2.93)

Thereforedud

= ut

+ cux

(2.94)

But, from (2.77), ut

+ cux

= 0. Therefore

dud

= 0 (2.95)


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Burgers equation, shock waves and expansion waves 19

The initial condition is u (x, 0) = u0 (x). Therefore

u (s, = 0) = u0 (s) (2.96)

Solving (3.19), we getu ( ) = k = constant (2.97)

Using (3.20), we obtain

u ( = 0) = u0 (s) = k or k = u0 (s) (2.98)

Thereforeu (s, ) = u0 (s) (2.99)

Since u = u, we obtainu (x, t ) = u0 (s) (2.100)

Using (3.13), we can write

u (x, t ) = u0 (x ct) = u (x ct,t = 0) (2.101)For a time interval t to t + t, we can write

u (x, t + t) = u (x c t, t ) (2.102)If we consider a point x j with neighbours x j 1 and x j +1 to the left and right sides of x j

00110011 0011

0011 0011

0 00 01 11 1 0011

00110 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

000000000000000000000000000000

111111111111111111111111111111

000000000000000000000000000000

111111111111111111111111111111

000000000000000000000000000000

111111111111111111111111111111

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

1 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 11 1 1 1 1 1

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

1 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 11 1 1 1 1 1 1

t t

t+ t

x x xx x j j+1 j1

c c t

c0


respectively, the foot of the characteristic can fall on the left or right side of x j dependingon the sign of c. Since the information travels a distance of x = ct along the characteristicduring a time t (from t = 0), c is the speed with which information propagates along thecharacterstics, and is called characteristic speed or wave speed . Therefore, we can seethat the sign of c determines the direction of information propagation, with informationcoming from the left if c > 0 and from right if c < 0.


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2.4 Burgers Equation, Shock Waves and ExpansionWaves

Consider the Burgers equation

ut

+ g (u)

x = 0, with g (u) =

12

u2 (2.103)

Let us check the wave speed for this case. The wave speed is dened by

a (u) = g (u)

u (2.104)

Therefore, a(u) = u. Consider an initial prole which is monotonically increasing. There-fore as u increases, a(u) increases.

x

u

t=0 t=t t=t 1 2

large speed

small speed

Figure 2.4: Formation of expansion waves for Burgers Equation

The larger values of u lead to larger speeds and smaller values of u lead to smallerspeeds. Therefore the upper parts of the prole move faster than the lower parts of theprole and the prole expands or gets rareed after some time. This phenomenon leadsto expansion waves or rarefaction waves .

Now, consider an initial prole which is monotonically decreasing, coupled with amonotonically increasing proe of the previous example. On the right part of the prolewhich is monotonically decreasing, as the upper part overtake lower part (due to largerspeed on the top), the gradient becomes innite and the solution becomes multi-valued.

Let us now recollect the basic features of a function. A function is rule that assignsexactly one real number to each number from a set of real numbers. Such a rule is oftengiven by an algebraic and/or trigonometric expression. A continuous function has nogaps or breaks at any point on its prole. Therefore, discontinuities are not allowed for acontinuous function.


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Shock waves in supersonic ows 21

u

x

t=0 t=t t=t t=t 1 2 3solutionunphysical

multivalued

Figure 2.5: Shock formation to avoid multivalued unphysical solution for Burgers equation

For the non-linear case and for an initial prole which is monotonically decreasing,the function may become multi-valued after some time. Then, the solution ceases to bea function, by denition. Discontinuities may appear and the solution becomes multi-valued. Multivalued functions are avoided on physical grounds. Imagine, for example,the density of a uid at a point having more than one value at any given time, which isunphysical. Therefore, as multi-valued functions are avoided, discontinuities will appear.These discontinuities are known as shock waves.

2.5 Shock Waves in Supersonic FlowsIn the supersonic ows of inviscid uid ows modeled by Euler equations, shock wavesappear when the ows are obstructed by solid bodies. The appearance of such shockwaves can be explained as follows.

Consider the ow of a uid over a blunt body, as shown in the following gure. The

Solid BodySubsonic flow(M < 1)

Figure 2.6: Subsonic ow over a blunt body


28/219



uid ow consists of moving and colliding molecules. Some of the molecules collide with

the solid body and get reected. Thus, there is a change in the momenta and energy of themolecules due to their collision with the solid body. The random motion of the moleculescommunicates this change in momenta and energy to other regions of the ow. At themacroscopic level, this can be explained as the propagation of pressure pulses. Thus,the information about the presence of the body will be propagated throughout the uid,including directly upstream of the ow, by sound waves. When the incoming uid owhas velocities which are smaller than the speed of the sound ( i.e. , the ow is subsonic),then the sound waves can travel upstream and the information about the presence of thesolid body will propagate upstream. This leads to the turning of the streamlines muchahead of the body, as shown in the gure (2.6).

Now, consider the situation in which the uid velocities are larger than the speed of the sound ( i.e. , the ow is supersonic). The information propagation by sound waves is

Solid Body

S oc Wave

Supersonic Flow M 1)

Figure 2.7: Supersonic ow over a blunt body with the formation of a shock wave

now not possible upstream of the ow. Therefore, the sound waves tend to coalesce ata short distance ahead of the body. This coalescence forms a thin wave, known as theshock wave, as shown in the gure (2.7). The information about the presence of the solidbody will not be available ahead of the shock wave and, therefore, the streamlines do not

change their direction till they reach the shock wave. Behind the shock wave, the owbecomes subsonic and the streamlines change their directions to suit the contours of thesolid body. Thus, the shock waves are formed when the a supersonic ow is obstructedby a solid body.

2.6 Mathematical Classication of PDEsWe can derive several simpler equations from the Navier-Stokes equations : pure con-vection equation, convection-diffusion equation, pure diffusion equation and the wave


29/219


First Order PDEs 23

equations of rst order and second order. These equations, including the Navier-Stokes

equations, are Partial Differential Equations (PDEs). To understand these equations bet-ter, we shall study their mathematical and physical behaviour. Let us start with theclassication of PDEs.

2.6.1 First Order PDEsGeneral form of a rst order linear PDE is

A(x, y)ux

+ B(x, y)uy

+ C (x, y)u = D(x, y) (2.105)

Let us simplify (2.105) by assuming C = D = 0. A(x, y)

ux

+ B(x, y)uy

= 0 (2.106)

The above equation is a rst order homogeneous partial differential equation (PDE). Letus look for the solutions of the form

u(x, y) = f (w) (2.107)

where w is some combination of x and y such that as x and y change, w remains constant.Substituting (2.107) in (2.106), we obtain

A(x, y)f (w)

wwx

+ B(x, y)f (w)

wwy

= 0

f (w)w

A(x, y)wx

+ B(x, y)wy

= 0 (2.108)

From the above equation, either df (w )dw = 0 or Adf dx + B

df dy = 0. Since we assumed that f is

a function of w only, df dw need not be zero. Therefore, the only possibility for (2.108) tobe true is to have

A(x, y)wx + B(x, y)

wy = 0 (2.109)

Let us now seek the solutions of (2.109) such that w remains constant as x and y vary.Therefore, require

dw = 0 or wx

dx + wy

dy = 0 (2.110)

From (2.109) and (2.110) (which look alike), we get

A(x, y)wx

= B(x, y)wy

(2.111)


30/219



andwx dx =

wy dy (2.112)

Dividing (2.112) by (2.111), we get

dxA(x, y)

= dyB(x, y)

(2.113)

Therefore, f(w) will be constant along those lines ( x, y) that satisfy (2.113). On integrating(2.113) for given A(x, y) and B(x, y), we get a functional relation between x and y, whichcan be taken as w. Thus, we can get

f (w) = f (w(x, y)) u(x, t ) = f (w) = f (w(x, y)) (2.114)

which will be the solution.Example:

ut

+ cux

= 0 (2.115)

A = 1 and B = c

dtA

= dxB

gives dt1

= dxc

dx = c dt (2.116)

Integrating, we getx = c t + k where k is a constant . (2.117)

w = k = x ct (2.118) u(x, t ) = f (w) = f (x ct) (2.119)

where f is an arbitrary function which must be determined by initial conditions for thePDE, which is (2.115) in this case. In a similar way, the non-homogeneous rst orderPDE, where C and D are non-zero, can also be solved [2].

2.6.2 CharacteristicsLet us rewrite the general form of rst order linear PDE (2.105) as

A(x, y)ux

+ B(x, y)uy

= C (x,y,u ) (2.120)


31/219


Characteristics 25

Let us now solve (2.120) for u(x, y) subject to the boundary condition

u(x, y) = (s) (2.121)

Let the boundary condition (2.121) be specied in the xy plane along a boundary curvewhich is described in parametric form asx = x(s) and y = y(s) (2.122)

Here, s in the arc length along the boundary. Along the boundary represented by (2.122),the variation of u is given by

du

ds =

u

x

dx

ds +

u

y

dy

ds (2.123)

Using (2.121), (2.123) can be written as

duds

= ux

dxds

+ uy

dyds

= dds

(2.124)

We now have two equations (2.120) and (2.124) with two unknowns ux

and uy

. We can

write (2.124) and (2.120) asux

dxds

+ uy

dyds

= dds

(2.125)

and ux

A + uy

B = C (2.126)

ordxds

dyds

A B

ux

uy

=

dds

C

(2.127)

We can solve (2.127) to obtain the unknowns ux

and uy

as

ux

uy

=

dxds

dyds

A B

1 dds

C

(2.128)

The solution is not possible if the determinant of the matrix, whose inverse is required,

is zero since M 1 = N T

|M | where N T is the transpose of the matrix N of cofactors of M .


32/219



The determinant is zero if dxds

dyds

A B

= 0 (2.129)

or

Bdxds A

dyds

= 0

Bdxds

= Adyds

dydsdxds

= B

A

dydx

= B

A

dxA

= dy

B (2.130)

We have already seen that (2.130) represents that curve in the ( x, y) plane in which w isa constant with ( u(x, y)) = f (w). Such curves are called characteristic curves or simplycharacteristics . Note also that the derivatives of the solution,

ux

or uy

may not exist

along the characteristics, since w = constant along the characteristics.

ux

= x

f (w) = f w

wx

= f w

0 = 0

uy

= y

f (w) = f w

wy

= f w

0 = 0

Therefore, discontinuities in solution may exist along the characteristics. That is why we

can solve for ux

and uy

everywhere in the (x, y) domain except along the characteristics

(when the determinant is zero (2.129)).

2.6.3 Second Order PDEsThe general form of a second order PDE is

A(x, y) 2ux 2

+ B(x, y) 2uxy

+ C (x, y) 2uy2

= D(x,y,u, ux

, uy

) (2.131)


33/219


Second order PDEs 27

Apart from the general form of the second order PDE, we can obtain two more relation-

ships by applying the chain rule to the total derivatives of ux and uy [3].

A 2ux 2

+ B 2uxy

+ C 2uy2

= D (2.132)

2ux 2

dx + 2uxy

dy = dux

(2.133)

2uxy

dx + 2uy2

dy = duy

(2.134)

or

A B C

dx dy 0

0 dx dy

2ux 2

2uxy

2uy2

=

D

dux

duy

(2.135)

The equation (2.135) can be solved for 2ux 2

, 2uxy

and 2uy2

everywhere in the (x, y)

domain, except on a curve where the determinant in (2.135) is zero, which will be the

characteristic curve. The zero determinant condition isA B C

dx dy 0

0 dx dy

= 0 (2.136)

A (dy)2 dx 0 B [dxdy 0] + C (dx)2 dy 0 = 0 (2.137) A (dy)2 B (dxdy) + C (dx)2 = 0 (2.138)

Let us divide by (2.138) by (dx)2 to obtain

Adydx

2

Bdydx

+ C = 0 (2.139)

This is the equation of the curve along which the second partial derivatives of u cannotbe dened. The solution to (2.139) is

dydx

= (B) (B)2 4AC 2A (2.140)dydx

= B (B)2 4AC 2A (2.141)


34/219


35/219


Physical signicance of the classication 29

Example 2 : Consider the pure diffusion equationut

= 2ux 2

2ux 2

= ut

A = , B = 0 & C = 0

B 2 4AC = 0 4 0 = 0

Therefore the equation is parabolic.

Example 3 : Consider the Laplace equation in 2-D 2ux 2

+ 2uy2

= 0 (2.146)

Here, A = 1, B = 0 and C = 1. B 2 4AC = 0 4 1 1 = 4. B 2 4AC < 0.This equation is elliptic.

2.6.4 Physical Signicance of the ClassicationThe mathematical classication introduced in the previous sections, leading to the cat-egorization of the equations of uid ows and heat transfer as hyperbolic, parabolic orelliptic, is signicant as different types of equations represent different physical behaviourand demands different types of treatment analytically and numerically.

Hyperbolic PDEs

Hyperbolic equations are characterized by information propagation along certain preferreddirections. These preferred directions are related to the characteristics of the PDEs.Consequently, there are domains of dependence and zones of inuence in the physicaldomains where the hyperbolic equations apply. The linear convection equation, the non-linear inviscid Burgers equation, the Euler equations, the inviscid isothermal equationsand the isentropic equations are all hyperbolic equations. As an illustration, let us considerthe wave equation (which describes linearized gas dynamics, i.e., acoustics), given by

2ut 2

= c2 2ux 2

(2.147)


36/219


37/219


Physical signicance of the classication 31

Parabolic PDEs

The parabolic equations are typically characterized by one direction information prop-agation. Unsteady heat conduction equation, unsteady viscous Burgers equation andunsteady linear convection-diffusion equation are examples of parabolic equations. Con-sider the unsteady heat conduction equation in 1-D, given by

T t

= 2T x 2

(2.152)

As done before, we can write this equation as

0 0dx dt 00 dx dt

2T x 2

2T xt

2T t 2

=

T t

dT x

dT t

(2.153)

Setting the determinant of the coefficient matrix to zero and solving for the slopes of thecharacteristic paths, we get

(dt)2 = 0dt = 0t = constant

(2.154)

Thus, there are two real but repeated roots associated with the characteristic equationfor unsteady conduction equation in 1-D. The characteristics are lines of constant time.The speed of information propagation (from the above equations) is

dxdt

= dx

0 = (2.155)

Thus, the information propagates at innite speed along the characteristics (lines of con-stant t). The domain of the solution for a typical parablic PDE (unsteady heat conductionequation) is shown in gure (2.9).

Elliptic PDEs

In contrast to the hyperbolic equations, the elliptic equations are characterized by infor-mation propagation having no preferred directions. Therefore, the information propagatesin all directions. A typical example is the steady state heat conduction in a slab. The


38/219



Figure 2.9: Domain of Solution for a Parabolic PDE (unsteady heat conduction equation)

pure diffusion equations in steady state are elliptic equations. Consider the steady heatconduction equation in 2-D, given by

2T x 2

+ 2T y2

= 0 (2.156)

As done before, the above equation can be written as

1 0 1dx dy 00 dx dy

2T x 2

2T xy

2T y2

=

0

dT x

dT y

(2.157)

Setting the determinant to zero and solving for the characteristics, we obtain

1 (dy)2 + 1 ( dx)2 = 0 or dydx

= 1 (2.158)Thus, the roots are complex and there are no real characteristics. That means, there areno preferred directions for information propagation. The domain of dependence and therange of inuence both cover the entire space considered. The domain of solution for atypical elliptic equation (steady state heat condution equation in 2-D) is shown in gure(2.10).


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Euler equations and hyperbolicity 33

Figure 2.10: Domain of Solution for an Elliptic PDE (steady heat conduction equation)

Not all equations can be classied neatly into hyperbolic, elliptic or parabolic equa-tions. Some equations show mixed behaviour. Steady Euler equations are hyperbolic forsupersonic ows (when Mach number is greater than unity) but are elliptic for subsonicows (when Mach number is less than unity). The mathematical behaviour of the uidow equations may change from one point to another point in the ow domain.

2.7 Euler equations and HyperbolicityWe have seen how a scalar equation (linear convection equation in this case) is hyperbolic,characterized by preferred directions of information propagation. Let us now consider thevector case for hyperbolicity.

Denition of hyperbolicity for systems of PDEs

Consider a system of PDEs U t

+ Gx

= 0 (2.159)

where

U =

U 1U 2...

U n

G =

G1G2...

Gn

(2.160)


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Here, U is the vector of conserved variables (also called as the eld vector) and G is the

vector of uxes (called as the ux vector), each component of which is a function of U .Usually, in uid dynamics, G is a non-linear function of U . Let us rewrite the abovesystem of equations in a form similar to the linear convection equation (in which the timeand space derivatives are present for the same conserved variable) as

U t

+ GU

U x

= 0 (2.161)

orU t

+ AU x

= 0 where A = GU

(2.162)

The above form of system of PDEs (2.162) is known as the quasi-linear form. Note that Awill be a nn matrix. A system of partial differential equations (2.162) is hyperbolic if the matrix A has real eigenvalues and a corresponding set of linearly independent eigenvectors.If the eigenvectors are also distinct, the system is said to be strictly hyperbolic . If thesystem is hyperbolic, then the matrix A can be diagonalised as

A = RDR 1 (2.163)

where R is the matrix of eigenvectors and D is the matrix of eigenvalues.

D =

1

0

0 0... ... ...0 n

, R = [R1, , Rn ] , ARi = iR i (2.164)

Therefore, we can dene a hyperbolic system of equations as a system with real eigenvaluesand diagonalisable coefficient (ux Jacobian) matrix.

Linear systems and characteristic variables

If A is constant, then the hyperbolic system U

t + A

U

x = 0 is linear. If we introduce a

characteristic variable asW = R1U (2.165)

then the hyperbolic system will be completely decoupled. If A is a constant, then so isR. Therefore

U t

= RW t

and U x

= RW x

(2.166)

ThereforeR

W t

+ ARW x

= 0 (2.167)


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Euler equations and hyperbolicity 35

orW t + R1AR =

W x = 0 (2.168)

ThereforeW t

+ DW x

= 0 (2.169)

1-D Euler equationsThe 1-D Euler equations

U t

+ U x

= 0 (2.170)

which can be written in quasi-linear form asU t

+ AU x

= 0 where A = GU

(2.171)

A =

0 1 0 3

2 u2 (3 ) u 1

22

u3 a2u 1

3 2 2

u2 + a2

1 u

(2.172)

In terms of the total enthalpy H = h + 12u2 = e + p + 12u

2 = p ( 1) + p + 12u

2

A =

0 1 0 3

2 u2 (3 ) u 1

u 1

2 u2 H H ( 1) u2 u

(2.173)

The eigenvalues of A are

1 = u

a, 2 = u and 3 = u + a (2.174)

and the corresponding eigenvectors are

R1 =1

u aH ua , R2 =

1u

12

u2 and R3 =

1u + a

H + ua (2.175)

Therefore, we can see that the 1-D Euler equations are (strictly) hyperbolic. So, aremulti-dimensional Euler equations.


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In this short course, apart from the traditional numerical methods for solving the

equations of compressible ows, alternative methodologies based on the Kinetic The-ory of Gases , called as Kinetic Schemes and a relatively new strategy of converting thenon-linear conservation equations into a linear set of equations known as the Relaxation Systems , along with the related numerical methods known as the Relaxation Schemes ,will be presented. The next two sections are devoted to the presentation of the governingequations for these two strategies.

2.8 Kinetic Theory, Boltzmann Equation and its Mo-ments as Macroscopic Equations

Consider the ow of air over a solid body, say a wing of an aeroplane. The variablesof interest are the uid velocities and the uid density, apart from the thermodynamicvariables like pressure and temperature, as they can be used to calculate the requireddesign parameters like lift, drag, thrust and heat transfer coefficients. To obtain thesevariables, we need to solve the Euler or Navier-Stokes equations, which is the subjectmatter of traditional CFD and some algorithms for doing so will be presented in thenext chapters. We can also consider the uid ow from a microscopic point of view,considering the ow of molecules and their collisions. Obviously, both the microscopicand the macroscopic approaches must be related, as we are referring to the same uid

ow. The macroscopic variables can be obtained as statistical averages of the microscopicquantities. This is the approach of the Kinetic Theory of Gases . Similar to the Navier-Stokes equations, which are obtained by applying Newtons laws of motion to the uids, wecan apply Newtons laws of motion to the molecules and, in principle, solve the resultingequations. But, it is practically impossible to solve the large number of equations thatresult, as there will be 1023 molecules in a mole of a gas. Neither can we know the initialconditions for all the molecules. Therefore, a better way of describing the uid ow at themicroscopic level is by taking statistical averages and the Kinetic Theory of Gases is basedon such a strategy. In the Kinetic Theory, the movement of the molecules is describedby probabilities instead of individual paths of molecules. The macroscopic quantities of

interest, like density, pressure and velocity, are obtained by taking statistical averagesof the molecular quantities. These averages are taken over macroscopically innitesimalbut microscopically large volumes. These averages are also known as moments and thisprocess of taking averages is called as taking moments .

Consider a small volume V ( V = 3r ) in the physical space (x,y,z). Let thenumber of molecules in this volume be 3N . Therefore, the local number density , whichrepresents the number of molecules per unit volume , is given by

n (r ) = Lim 3 r 0 3N 3r

(2.176)


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Kinetic theory of gases 37

Therefore, we can write

d3N = n (r ) d

3r (2.177)

and we can calculate the number of molecules if the local number density (or the molec-ular distribution) is known. If we consider a phase space , which has three additionalcoordinates as the molecular velocities apart from three physical coordinates, we canwrite

d6N = f p (r , v) d3rd 3v = f p (r , v ) dxdydz dv1dv2dv3 (2.178)

where f p(r , v) is the local number density in the phase space, known as the phase space distribution function . Therefore, if the phase space distribution function is known, we cancalculate the number of molecules by integrating the phase space distribution function

(thereby obtaining the physical number density) as

n (r ) = f p (r , v) d3v (2.179)which we denote by a simpler notation as

n = f p (2.180)

Multiplying both sides of the above equation by the mass of the molecules, m, and iden-tifying the density of the gas as the number of molecules multiplied by the mass of the

molecules, we obtainmn = = mf p = f where f = mf p (2.181)

Similarly, the average or mean speed of the molecules can be written as

n (r ) v = vf p (r , v) d3v (2.182)or

n v = vf p (2.183)

Multiplying by the mass, we obtain

nm v = vmf p (2.184)

or v = v f (2.185)

Denoting the average molecular velocity v by u and recognizing it as the uid velocity,we can writeu = vf (2.186)


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Similarly, we can obtain an average for the kinetic energy as

E =12

v2f (2.187)

Thus, we have the expressions

= f ; u = v f ; E =12

v2f (2.188)

which give the macroscopic quantities as averages (also called as moments) of the molec-ular velocity distribution function. In addition to the above moments, we can also derivethe following additional moments for the Pressure tensor and the heat ux vector.

P ij = p ij ij = cic j f where c = v u (2.189)and

q i = cci f (2.190)

The relative velocity c is known by various names as peculiar velocity , random velocity orthermal velocity . Here, ij is the Kronecker delta function, dened by

ij = 1 if i = j0 if i = j (2.191)

andc2 = c21 + c

22 + c

23 ; v

2 = v21 + v22 + v

23 (2.192)

The expression for E can be evaluated as

E = p2

+ u2

2 (2.193)

But, the right expression for E is

E = e + u2

2 (2.194)

wheree = p

( 1) (2.195)

is the internal energy. Therefore, to get the right value of E , we have to modify themoment denitions. But, rst let us learn about the equilibrium distribution. If wekeep a system isolated from the surroundings and insulated (no heat transfer), and if there are no internal heat sources and external forces, the gas in the system will reachthermodynamic equilibrium . The velocity distribution of such a state is known as theequilibrium distribution. It is also known as Maxwellian distribution . In such a state,all gradients are zero (the gas is at rest). However, the ows of interest always contain


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We rst need to evaluate the fundamental integral

e

x 2 dx. Let

K = ex 2 dx = ey2 dy (2.207)Since denite integral is a function of limits only, we get

K 2 = ex 2 ey2 dxdy = e(x 2 + y2 )dxdy (2.208)Let x = r cos and y = r sin . Then

K 2

= 0

2

0 er 2

rdrd

K 2 = []20 0 er 2 rdrK 2 = 2

12

er 20

=

K = Therefore,

ex 2 dx = (2.209)

Using the above, we can derive the following expressions.

J 0 = J +0 = 2J 1 = 0 J +1 =

12

J 2 = 2

J +2 = 4

J 3 = 0 J +3 = 12

J 4 = 3

4 J +4 = 3

8

(2.210)

and J n = J n J +n (2.211)Using these integrals, if we derive U 3, we get, for a Maxwellian

U 3 =v2

2 F =

p2

+ 12

u2 = E (2.212)

where E = p2

+ u2

2 (2.213)


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But, for Euler equations, we know that

E = p

( 1) +

u2

2 (2.214)

This discrepancy is because we considered only a mono-atomic gas which has no internaldegrees of freedom contributing to internal energy. It has only translational degrees of freedom. A polyatomic gas has internal degrees of freedom contributing to vibrational androtational energies. To add internal energy contributions, we modify the denitions asfollows.

= 0 dI

d3

v f (2.215)

u = 0 dI d3v vf (2.216)E = 0 dI d3v (I + v

2

2 )f (2.217)

where v2 = v21 + v22 + v

23 (2.218)

Here I is the internal energy variable corresponding to non-translational degrees of free-dom. The Maxwellian is modied as

F = I 0

D2

e (v u )2 e I

I 0 (2.219)

where I 0 = (2 + D) D

2 ( 1) RT (2.220)

is the average internal energy due to non-translational degrees of freedom. The basicequation of kinetic theory is the Boltzmann equation

f

t + v

f

x = J (f, f ) (2.221)

(in the absence of external forces).The left hand side of (2.221) represents the temporal and spatial evolution of the

velocity distribution function, f . The right hand side, J (f, f ) represents the collision term.The molecules are in free ow except while undergoing collisions. The LHS represents thefree ow and the RHS represents the changes in the velocity distribution due to collisions.The collision term makes (2.221) an integro-differential equation, which is difficult tosolve. For an introduction to the Kinetic Theory of Gases, the reader is referred to thefollowing books [4, 5, 6].


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2.8.1 B-G-K Model for the collision termBhatnagar, Gross and Krook [7] proposed a simple model for the collision term :

J (f, f ) = F f

tR(2.222)

According to the B-G-K model, the velocity distribution function, f , relaxes to a Maxwelliandistribution, F , in a small relaxation time, tR . With this model, the Boltzmann equationbecomes

f t

+ vf x

= F f

tR(2.223)

2.8.2 Splitting MethodBoltzmann equation (2.221) is usually solved by a splitting method. To illustrate thesplitting method, let us consider the (2.223). We can re-write (2.223) as

f t

= vf x

+ F f

tR(2.224)

f t

= O1 + O2 (2.225)

where O1 = vf x

(2.226)

and O2 = F f

tR(2.227)

We can split (2.225) into two steps:

f

t = O1 (2.228)

f t

= O2 (2.229)

(2.228) and (2.229) can be re-written as

f t

+ vf x

= 0 (2.230)


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f

t =

F f tR

(2.231)

equation (2.230) is the convection equation for f and equation (2.231) is the collisionequation for f . Therefore, The operator splitting has resulted in two steps: (i) a convectionstep and (ii) a collision step. In the convection step, (2.230) can be solved exactly ornumerically. Let us see how, in the collision step, the equation (2.231) can be solved. Wecan write (2.231) as an ODE.

df dt

= F f

tR(2.232)

This is a simple ODE for which the solution is given by

f = ( f 0 F ) e t

t R + F (2.233)

or f = f 0e tt R + F 1 e

tt R (2.234)

If we take tR = 0, then (2.233) gives

f = f 0et0 + F or f = F (2.235)

Therefore, If the relaxation time is zero, the exact solution of the collision step drives thedistribution to a Maxwellian. Thus, the collision step becomes a relaxation step . The

kinetic schemes or Boltzmann schemes are based on this split-up into a convection stepand a relaxation step :Convection Step :

f t

+ vf x

= 0 (2.236)

Relaxation Step :f = F (2.237)

Therefore, in kinetic schemes, after the convection step, the distribution function instan-taneously relaxes to a Maxwellian distribution. If we start with an initially Maxweliandistribution, we can use the approximation

F t

+ v F x

= 0 (2.238)

as a basis for developing kinetic schemes. The Euler equations

Ut

+ G ix i

= 0 ( i = 1, 2, 3) (2.239)

where U =

u iE

(2.240)


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G =

u i

ij p + ui u j pui + ui E (2.241)

E = p ( 1)

+ u2

2 (2.242)

and u2 = u21 + u22 + u23 (2.243)

can be obtained as moments of the Boltzmann equation:1

viI +

v2

2

f t + vi

f x i = 0 (2.244)

Therefore , U =

1vi

I + v 2

2

f = 0 dI d3v1vi

I + v2

2

f (2.245)

and G j =

1vi

I + v2

2

v j f = 0 dI d3v1vi

I + v2

2

v j f (2.246)

The Kinetic Schemes are based on the above connection between the Boltzmann equationand Euler equations. The splitting method is also an inherent part in most of the KineticSchemes.

2.9 Relaxation Systems for Non-linear ConservationLaws

In the previous section, the non-linear vector conservation laws of Fluid Dynamics werederived from a simpler linear convection equation (the Boltzmann equation), using theKinetic Theory of Gases. Thus, the task of solving the non-linear vector conservationequations was simplied by the use of a linear convection equation. In this section,another such a simpler framework is presented, in which the non-linear conservation lawsare linearized by a Relaxation Approximation . This framework of a Relaxation System iseven simpler than the previous one and is easier to deal with.

Consider a scalar conservation law in one dimensionut

+ g (u)

x = 0

with the initial condition u (x, t = 0) = u0 (x) .(2.247)


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Relaxation systems for non-linear conservation laws 45

Here the ux g (u) is a non-linear function of the dependent variable u. With g (u) = u2

2 ,

we recover the inviscid Burgers equation. The main difficulty in solving this equationnumerically is the non-linearity of the ux g (u). Jin and Xin [8] dealt with this problemof non-linearity by introducing a new variable v, which is not an explicit function of thedependent variable u and provided the following system of equations.

ut

+ vt

= 0

vt

+ 2ux

= 1 [v g (u)]

(2.248)

Here, is a positive constant and is a very small number approaching zero. We canrearrange the second equation of the above system (2.248) as

vt

+ 2ux

= [v g (u)] (2.249)and as 0, we obtain v = g(u). Substituting this expression in the rst equationof the Relaxation System (2.248), we recover the original non-linear conservation law(2.247). Therefore, in the limit 0, solving the Relaxation System (2.248) is equivalentto solving the original conservation law (2.247). It is advantageous to work with theRelaxation System instead of the original conservation law as the convection terms arenot non-linear any more. The source term is still non-linear, and this can be handledeasily by the method of splitting. The initial condition for the new variable v is given by

v(x, t = 0) = g (u0 (x)) (2.250)

This initial condition avoids the development of an initial layer, as the initial state is inlocal equilibrium [8]. The above approach of replacing the non-linear conservation law bya semi-linear Relaxation System can be easily extended to vector conservation laws andto multi-dimensions. Consider a vector conservation law in one dimension, given by

Ut

+ G (U )

x = 0 (2.251)

Here, U is the vector of conserved variables and G (U ) is the ux vector, dened by

U =

uE

and G (U ) =u

p + u2 pu + uE

(2.252)

where is the density, u is the velocity, p is the pressure and E is the total internal energyof the uid, dened by

E = p

( 1) +

u2

2 (2.253)


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with being the ratio of specic heats. The above vector conservation laws are the Euler

equations of gas dynamics and describe the mass, momentum and energy conservationlaws for the case of an inviscid compressible uid ow. The Relaxation System for theabove vector conservation laws is given by

Ut

+ Vx

= 0

Vt

+ D Ux

= 1 [V G (U )]

(2.254)

where D is a positive constant diagonal matrix, dened by

D =D1 0 00 D2 00 0 D3

(2.255)

The positive constants in the Relaxation System for the scalar case (2.248) and D i , (i =1, 2, 3) in the Relaxation System for the vector case (2.254) are chosen in such a way thatthe Relaxation System is a dissipative (stable) approximation to the original non-linearconservation laws. To understand this better, let us do a Chapman-Enskog type expansionfor the Relaxation System .

2.9.1 Chapman-Enskog type expansion for the Relaxation Sys-tem

In this section, a Chapman-Enskog type expansion is performed for the Relaxation System ,following Jin and Xin [8]. We can rewrite the second equation of the Relaxation System (2.248) as

v = g (u) vt

+ 2ux

(2.256)

which means thatv = g (u) + O [ ] (2.257)

Differentiating with respect to time, we obtain

vt

= t

[g (u)] + O [ ] = gu

ut

+ O [ ] (2.258)

Since the rst equation of the Relaxation System (2.248) gives

ut

= vx

(2.259)

we can writevt

= gu

vx

+ O [ ] (2.260)


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Therefore, using (2.257), we can write

vt

= gu

x {g (u) + O [ ]} + O [ ] (2.261)

or vt

= gu

gu

ux

+ O [ ] = gu

2 ux

+ O [ ] (2.262)

Substituting the above expression in (2.256), we get

v = g (u) gu

2 ux

+ O [ ] + 2ux

(2.263)

or

v = g (u) ux

2 gu

2

+ O 2 (2.264)

Substituting this expression for v in the rst equation of the Relaxation System (2.248),we get

ut

+ g (u)

x =

x

ux

2 gu

2

+ O 2 (2.265)

The right hand side of (2.265) contains a second derivative of u and hence represents a dis-sipation (viscous) term. The coefficient represents the coefficient of viscosity. Therefore,the Relaxation System provides a vanishing viscosity model to the original conservationlaw. For the coefficient of dissipation to be positive (then the model is stable), the fol-lowing condition should be satised.

2 gu

2

or gu (2.266)

This is referred to as the sub-characteristic condition . The constant in the Relaxation System (2.248) should be chosen in such a way that the condition (2.266) is satised.

For the vector conservation laws (2.251) modeled by the Relaxation System (2.254),the ChapmanEnskog type expansion gives

Ut

+ G (U )

x =

x

D G (U )

U

2 Ux

+ O( 2) (2.267)

For the Relaxation System (2.254) to be dissipative, the following condition should besatised.

D G (U )

U

2

0 (2.268)


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Based on the eigenvalues of the original conservation laws (2.251), i.e., Euler equations,

Jin and Xin [8] proposed the following two choices.

(i) Dene D as D =21 0 00 22 00 0 23

First choice : 2 = 21 = 22 =

23 = max [|u a|, |u|, |u + a|] (2.269)

(ii ) Second choice : 21 = max |u a|, 22 = max |u| and 23 = max |u + a| (2.270)where u is the uid velocity and a is the speed of sound. With the rst choice, the diagonalmatrix D can be written as

D = 2

I (2.271)where I is a unit matrix.

2.9.2 Diagonal form of the Relaxation SystemThe Relaxation System (2.248) can be written in matrix form as

Qt

+ A Qx

= H (2.272)

where Q = uv , A =

0 12 0 and H =

0

1 [v g (u)] (2.273)As the Relaxation System (2.248) is hyperbolic, so is (2.272) and, therefore, we can write

A = RR1 and consequently = R1AR (2.274)

where R is the matrix of right eigenvectors of A, R1 is its inverse and is a diagonalmatrix with eigenvalues of A as its elements. The expressions for R, R1 and are givenby

R = 1 1

, R1 =

12

12

12

12

and =

0

0 (2.275)

Since the Relaxation System (2.272) is a set of coupled hyperbolic equations, we candecouple it by introducing the characteristic variables as

f = R1Q which gives Q = Rf (2.276)

Therefore, we can write Qt

= R f t

and Qx

= R f x

(2.277)


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Substituting the above expressions in (2.272), we obtain

f t

+ R1AR f x

= R1H (2.278)

Using (2.274), the above equation can be written as

f t

+ f x

= R1H (2.279)

where

f = f 1f 2 = R1Q =

u2 v2u2

+ v2

and R1H =12 [v g (u)]

12

[v g (u)](2.280)

Thus, we obtain two decoupled equations as

f 1t

f 1x

= 12

[v g (u)]f 2t +

f 2x =

12 [v g (u)]

(2.281)

Solving these two equations in the limit of 0 is equivalent to solving the originalnon-linear conservation law (2.247). It is much easier to solve the above two equationsthan solving (2.247), since the convection terms in them are linear. The source terms arestill non-linear, but these can be handled easily by the splitting method, which will bedescribed in the following sections. Using (2.275) and (2.276), we obtain the expressions

u = f 1 + f 2 and v = (f 2 f 1) (2.282)

using which we can recover the original variables u and v. In the case of vector conservationlaws (2.251), the diagonal form of the Relaxation System leads to

f 1t

f 1x

= 12

[V G (U )] f 2t

+ f 2x

= 12

[V G (U )](2.283)

where f 1 and f 2 are vectors with three components each for the 1-D case.


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2.9.3 Diagonal form as a Discrete Kinetic SystemThe diagonal form of the Relaxation System can be interpreted as a discrete Boltzmannequation [9, 10, 11]. Let us introduce a new variable F as

F = F 1F 2 =

u2

g (u)2

u2

+ g(u)

2

(2.284)

With these new variables, the diagonal form of the Relaxation System (2.279) can berewritten as

f t

+ f x

= 1 [F f ] (2.285)This equation is similar to the Boltzmann equation of Kinetic Theory of Gases with aBhatnagar-Gross-Krook (B-G-K) collision model, except that the molecular velocities arediscrete ( and ) and the distribution function f correspondingly has two components,f 1 andf 2. The new variable F represents the local Maxwellian distribution. This interpre-tation was used by Natalini [9] and Driollet & Natalini [10] to develop multi-dimensional Relaxation Systems which are diagonalizable and new schmes based on them. The classicalBoltzmann equation with B-G-K model in one dimension is given by

f t

+ f x

= 1tR

[F f ] (2.286)

where is the molecular velocity (we are not using v as it has been used in the Relax-ation System for the new variable), tR is the relaxation time and F is the equilibrium(Maxwellian) distribution. The Euler equations can be obtained as moments of the Boltz-mann equation. The 1-D local Maxwellian for such a case is given by

F = I 0

12

e (u )2 + I I 0 (2.287)

where is the density, D is the number of translational degrees of freedom, = 12RT ,T is the temperature, I is the internal energy variable for the non-translational degreesof freedom and I 0 is the corresponding average internal energy. The moments of thedistribution function lead to the macroscopic variables as

u = 0 dI d 1

I + 2

2

f = 0 dI d 1

I + 2

2

F (2.288)


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and

g (u) = 0 dI d 1

I + 2

2

f = 0 dI d 1

I + 2

2

F (2.289)

The macroscopic equations (Euler equations in this case) are obtained as moments of theBoltzmann equation by

0 dI d 1

I + 12

2f t

+ f x

= 1tR

[F f ] (2.290)

The corresponding expressions for the moments of the discrete Boltzmann equation areu = P f = P F , v = P f and g (u) = P F where P = [1 1] (2.291)

for the case of scalar conservation laws and

U = P f = P F , V = P f and G (U ) = P F (2.292)

for the case of vector conservation laws. The macroscopic equations are obtained from thediscrete Boltzmann equation by multiplying by P and P respectively. Let us multiplythe discrete Boltzmann equation (2.285) by P to obtain

P f

t +

f

x = P

1 [F

f ] (2.293)

or (P f )

t +

(P f )x

= 1

[P F P f ] (2.294)Using (2.291), the above equation can be rewritten as

ut

+ vx

= 0 (2.295)

which is the rst equation of the Relaxation System (2.248). Similarly, multiplying thediscrete Boltzmann equation (2.285) by P , we obtain

(P f )

t +

(P 2f )

x =

1 [P F

P f ] (2.296)

Evaluating P 2f as 2u and using (2.291), we getvt

+ 2ux

= 1 [v g (u)] (2.297)

which is the second equation of the Relaxation System (2.248). The Relaxation System forthe vector conservation laws can also be recovered by a similar procedure. In comparisonwith the classical Boltzmann equation, we can see that recovering the moments are simplerfor the Relaxation System and therefore the Relaxation Schemes will be simpler than thetraditional Kinetic Schemes in nal expressions.


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2.9.4 Multi-dimensional Relaxation SystemsConsider a scalar conservation law in 2-D

ut

+ g1 (u)

x +

g2 (u)y

= 0 (2.298)

The Relaxation System given by Jin and Xin [8] for the above equation is

ut

+ v1x

+ v2y

= 0

v1t

+ 21ux

= 1 [v1 g1 (u)]

v2t + 22 uy = 1 [v2 g2 (u)]

(2.299)

We can write the above Relaxation System in matrix form as

Qt

+ A1 Qx

+ A2 Qy

= H (2.300)

where

Q =uv1v2

, A1 =0 1 021 0 00 0 0

, A2 =0 0 10 0 022 0 0

and H =

0

1 {v1 g1 (u)}

1 {v2 g2 (u)}

(2.301)

The matrices A1 and A2 do not commute (A1A2 = A2A1) and the above system is notdiagonalizable. This is true in general for the multi-dimensional Relaxation System of Jinand Xin (see [9]). As it is preferable to work with a diagonal form, Driollet and Natalini[10] generalize the discrete Boltzmann equation in 1-D to multi-dimensions to obtain amulti-dimensional Relaxation System as

f t +D

k=1k f x k

= 1 [F f ] (2.302)For the multi-dimensional diagonal Relaxation System , the local Maxwellians are denedby [10]

F D +1 = 1D

u + 1

D

k=1

gk (u)

F i = 1

gi (u) + F D +1 , (i = 1, , D )(2.303)


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Let us consider the 2-D case for which the local Maxwellians are given by

F =F 1F 2F 3

=

u3

23

g1 (u) + 13

g2 (u)

u3

+ 13

g1 (u) 23

g2 (u)

u3

+ 13

g1 (u) + 13

g2 (u)

(2.304)

Using the denitions

u = P f = P F , g1 = P 1F , g2 = P 2F , v1 = P 1f and v2 = P 2f (2.305)

we can o

computational gas dynamics

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