Closure problem

Closure problem and turbulence modeling Because turbulence consists of random fluctuations of the various flow properties, we use a statistical approach. Our purposes are best served by using the procedure introduced by Reynolds (1895) in which all quantities are expressed as the sum of mean and fluctuating parts(Reynolds decomposition). We then form the time average of the continuity and Navier-Stokes equations. The result of using the Reynolds decomposition in the NS equations is called the RANS or Reynolds Averaged Navier Stokes Equations. Upon substitution of the Reynolds decomposition (for each variable, we substitute the corresponding decomposition) we obtain the following RANS equations :

The nonlinearity of the Navier-Stokes equation leads to theappearance of momentum fluxes

that act as apparent stresses throughout the flow which called Reynolds stress tensors −ρ ui u j

These momentum fluxes are unknown a priori.

w h ere ;R ı j=−ρ ui u j=−ρ∗[ uu uvvuwu

vvwv

uwvwww ]

IfRı j is symmetric, then Rı j=R ji and there are six independent components, instead of nine

w h ere ;R ı j=−ρ∗[ uuvvww

uwvuwv ]

Then we need a model for ui u j to close the equation system. This is called the closure problem: the number of unknowns (ten: three velocity components, pressure, six stresses) is larger than the number of equation (four: the continuity equation and three components of the Navier stokes equations).This illustrates the issue of closure which establishinga sufficient number of equations for all of the unknowns.One of the methods to solve the closure problem is the use of turbulence models.Turbulence modeling

A turbulence model is a computational procedure to close the system of mean flow equations.

Define the Reynolds stresses in terms on known (averaged) quantities For most engineering applications it is unnecessary to resolve the details of the turbulent

fluctuations. Turbulence models allow the calculation of the mean flow without first calculating the

full time-dependent flow field. We only need to know how turbulence affected the mean flow. In particular we need expressions for the Reynolds stresses.

For a turbulence model to be useful in a general purpose CFD code it must have wide applicability, be accurate, simple and economical to run.

The most common turbulence models are classified here;Classification of turbulent models Nowadays turbulent flows may be computed using several different approaches. Either by solving the Reynolds-averaged Navier-Stokes equationswith suitable models for turbulent quantities or by computing them directly. The main approaches are summarized below.Reynolds-Averaged Navier-Stokes (RANS) Models • Eddy-viscosity models (EVM) One assumes that the turbulent stress is proportional to the mean rate of strain. Furthermore eddy viscosity is derived from turbulent transport equations (usually k + one other quantity). • Non-linear eddy-viscosity models (NLEVM) Turbulent stress is modeled as a non-linear function of mean velocity gradients. Turbulent scales are determined by solving transport equations (usually k + one other quantity). Model is set to mimic response of turbulence to certain important types of strain. • Differential stress models (DSM) This category consists of Reynolds-stress transport models (RSTM) or second-order closure models (SOC). One is required to solve transport equations for all turbulent stresses

(u' )2 , ( v ' )2 , (w' )2 , (u ' v ' ) , (u' w' ) , (v ' w' ) ,and this will increase the number of equations to solve by six.Computation of fluctuating quantities • Large-eddy simulation (LES) Based on space-filtered equations. Time dependent calculations are performed. Large eddies are explicitly calculated. For small eddies, their effect on the flow pattern is taken into account with a “sub-grid model” of which many styles are available.• Direct numerical simulation (DNS) No modeling what so ever is applied. One is required to resolve the smallest scales of the flow as well. Extend of modeling for certain CFD approach is illustrated in the following figure. It is clearly seen, that models computing fluctuation quantities resolve shorter length scales than models solving RANS equations. Hence they have the ability to provide better results. However they have a demand of much greater computer power than those models applying RANS methods.

Brief History of Turbulence ModelingThe origin of the time-averaged Navier-Stokes equations dates back to the late nineteenth centurywhenReynolds (1895)published results from his research on turbulence. The earliest attempts at developing a mathematical description of the turbulent stresses, which is the core of the closure problem, were performed by Boussinesq (1877) with the introduction of the eddy viscosityconcept. Neither of these authors, however, attempted to solve the time-averaged Navier-Stokes equations in any kind of systematic manner.More information regarding the physics of viscous flow was still required, until Prandtl's discovery of the boundary layer in 1904.Prandtl (1925) later introduced the concept of the mixing-length model, which prescribed an algebraic relation for the turbulent stresses. This early development was the cornerstone for nearly all turbulence modeling efforts for the next twenty years. The mixing length model is now known as an algebraic or zero-equation model.To develop a more realistic mathematical model of the turbulent stresses, Prandtl (1945) introduced the first one-equation model by proposing that the eddy viscosity depends on the turbulent kinetic energy, k, solving a differential equation to approximate the exact equation for k. This one equation model improved the turbulence predictions by taking into account the effects of flow history.The problem of specifying a turbulence length scale still remained. This information, which can be thought of as a characteristic scale of the turbulent eddies, changes for different flows, and thus is required for a more complete description of the turbulence.A more complete model would be one that can be applied to a given turbulent flow by prescribing boundary and/or initial conditions.Kolmogorov (1942) introduced the first complete turbulence model, by modeling the turbulent kinetic energy k, and introducing a second parameter ω that he referred to as the rate of dissipation of energy per unit volume and time.This two-equation model, termed the k-ωmodel, used the reciprocal of ω as the turbulence time

scale, while the quantity k0.5

ω served as a turbulence length scale, solving a differential equation

for ω similar to the solution method for k.Because of the complexity of the mathematics, which required the solution of nonlinear differential equations, it went virtually without application for many years, before the availability of computers.Rotta (1951) pioneered the use of the Boussinesq approximation in turbulence models to solve for the Reynolds stresses. This approach is called a second-order or second-moment closure.Such models naturally incorporate non-local and history effects, such as streamline curvature andbody forces. The previous eddy viscosity models failed to account for such effects.For a three dimensional flow, these second-order closure models introduce seven equations, one for aturbulence length scale, and six for the Reynolds stresses. As with Kolmogorov's k-ω

model, theComplex nature of this model awaited adequate computer resources.

Thus, by the early 1950's, four main categories of turbulence models had developed:(1) Algebraic (Zero-Equation) Models(2) One-Equation Models(3) Two-Equation Models(4) Second-Order Closure ModelsWith increased computer capabilities beginning in the 1960's, further development of all four ofthese classes of turbulence models has occurred.The most important modern developments aregiven below for each class:Algebraic (Zero-Equation) ModelsVan Driest (1956) devised a viscous damping correction for the mixing-length model. This correction is still in use in most modern turbulence models.Cebeci and Smith (1974) refined theeddy viscosity/mixing-length concept for better use with attached boundary layers.Baldwin andLomax (1978) proposed an alternative algebraic model to eliminate some of the difficulty indefining a turbulence length scale from the shear-layer thickness.One-Equation ModelsWhile employing a much simpler approach than two-equation or second-order closure models one-equation models have been somewhat unpopular and have not showed a great deal ofsuccess.One notable exception was the model formulated by Bradshaw, Ferris, and Atwell(1967), whose model was tested against the best experimental data of the day at the 1968Stanford Conference on Computation and Turbulent Boundary Layers.There has been somerenewed interest in the last several years due to the ease with which one-equation models can beSolved numerically, relative to more complex two-equation or second-order closure models.Two-Equation ModelsWhile Kolmogorov's k-ω model was the first two-equation model, the most extensive work hasbeen done by Daly and Harlow (1970)and Launder and Spalding (1972). Launder's k-ε

model is the most widely used two-equation turbulence model; here εis the dissipation rate of turbulentkinetic energy. Independently of Kolmogorov, Saffman (1970) developed a k-ωmodel thatshows advantages to the more well-known k-ε model, especially for integrating through

theviscous sub-layer and in flows with adverse pressure gradients.Due to the increased complexity of this class of turbulence models, second-order closure modelsdo not share the same wide use as the more popular two-equation or algebraic models. The mostnoteworthy efforts in the development of this class of models was performed by Donaldson andRosenbaum (1968), Daly and Harlow (1970), and Launder, Reece, and Rodi (1975). The latterhas become the baseline

second-order closure model, with more recent contributions made byLumley (1978), Speziale (1985), Reynolds (1987), and many other thereafter, who haveadded mathematical rigor to the model formulation.While the present study is not intended to be a complete catalogue of all turbulence modelsCommon turbulence modelsThe turbulent models are as follows, in order of increasing complexity:1. Algebraic Zero equation model: mixing length model.2. One equation model: Spalart-Almaras k‐ model, μt‐model 3. Two equation models: k- style models (standard, RNG, realizable), k- model.4. �Algebraic stress models: ASM (second order model)5. Seven equation model: Reynolds stress model(RSM) .

• The number of equations denotes the number of additional PDEs that are being solved.FIRST ORDER MODELSFirst order models are based on the analogy between laminar and turbulent flow. They are also called Eddy Viscosity Models (EVM). The idea is that there exists an analogy between the action of viscous stressesτ ij and Reynolds stresses−ρ u 'i u' j on the mean flow. Both stresses appear on the right hand side of the momentum equation and in Newton's law of viscosity the viscous stresses are taken to be proportional to the rate of deformation of fluid elements(e ij).

For an incompressible fluid this givesτ ij=μ e ij=μ(∂ ui

∂ x j

+∂ u j

∂ xi

)

In order to simplify the notation using the suffix notation where i, j, and k denote the x, y, and z

directions respectively, viscous stresses are given by:τ ij=μ e ij=μ( ∂ ui

∂ x j

+∂ u j

∂ xi)=μ( ∂ u

∂ y+ ∂ v

∂ x)

It is experimentally observed that turbulence decays unless there is shear inisothermal incompressible flows. Furthermore, turbulent stresses are found to increase as the mean rate of deformation increases. It was proposed by Boussinesq in 1877 that Reynolds stresses could be linked to mean rates of deformation. Using thesuffix notation we get

τ ij=− ρu 'i u

'j=μ t(

∂ U i

∂ X j

+∂ U j

∂ X i

)

The right hand side is analogous to this formula

τ xy=τ yx=μ ( ∂ u∂ y

+ ∂ v∂ x

)Except for the appearance of the turbulent or eddy viscosity μt(dimensions Pa

s). There is also a kinematic turbulent or eddy viscosity denoted by ν t=μt

ρ with dimensionsm2

s.

The turbulent viscosity is not homogeneous, it varies in space It is, however assumed to be isotropic, in other words that the ratio between Reynolds stress and mean rate ofdeformation is the same in all directions. This assumption is valid for many flows, but not for all (e.g. flows with strong separation or swirl).

ZERO EQUATION MODELSIn zero equation models, as the name designates, we have no PDE that describes the transport of the turbulent stresses and fluxes. A simple algebraic relation is used to close the problem.

Based on the mixing length theory, which is the length conceptually analogous to the concept of mean free path in thermodynamics: a fluid parcel will conserve its properties for a characteristic length,l , before mixing with the surrounding fluid.On dimensional grounds we assume that the kinematic turbulent viscosityν twhichhas

dimensions m2

scan be expressed as a product of a turbulent velocity scale ϑ(m/s) and a length

scale l (m). If one velocity scale and one length scale suffice to describe the effects of turbulence dimensional analysis yields ν t ¿we then assume that the velocity scale is proportional to the length scale and the gradients in the

velocity (shear rate, which has dimension 1/s):ϑ∝l|∂ U∂ y |

we can derive Prandtl’s (1925) mixing length model:ν t=lm2|∂U

∂ y |wherelm is determined experimentally .Algebraic expressions exist for the mixing length for simple two-dimensionalflows, such as pipe and channel flow.

Mixing length model assessment• Advantages:

– Easy to implement.– Fast calculation times.– Good predictions for simple flows where experimental correlations for the mixing

length exist.• Disadvantages:

– Completely incapable of describing flows where the turbulent length scale varies: anything with separation or circulation.

– Only calculates mean flow properties and turbulent shear stress.• Use:

– Sometimes used for simple external aero flows.– Pretty much completely ignored in commercial CFD programs today.

• Much better models are available.ONE EQUATION MODELSIn one equation models a transport equation (PDE )is often solved for the turbulent kinetic energ (per unit mass) k.The unknown turbulent length scale must be given. This length scale is, for example, taken as proportional to the thickness of the boundary layer, the width of a jet or a wake.And the unknowns (turbulent viscosity) are expressed as a function of the turbulent kinetic energy as:

k=12(u2+v2+w2)

The instantaneous kinetic energy k(t) of a turbulent flow is the sum of mean kinetic energy K and turbulent kinetic energyk :

K t=K+k

K=12(U 2+V 2+W 2)

We also make use of the fact that  ν t∝ϑ l but in this case , the velocity scale is proportional to the square root of the kinetic energy (unlike the above case where ϑ  was proportional to the gradient of vel-ocity). Therefore, we have:ν t=lm √k

Now that the turbulent viscosity is expressed in terms of the turbulent kinetic energy (therefore the turbulent stresses is functions of the kinetic energy), a PDE is developed for the mean kinetic energy.

Where;Eij is the mean rate of deformation tensor.And This equation can be read as:

– (I) the rate of change of K, plus– (II) transport of K by convection, equals– (III) transport of K by pressure, plus– (IV) transport of K by viscous stresses, plus– (V) transport of K by Reynolds stresses, minus– (VI) rate of dissipation of K(ε k), minus– (VII) Turbulence production.

And the PDE for Turbulent kinetic energyis as follows:

Where eij’ is fluctuating component of rate of deformation tensor.On the other hand, this equation introduces two new unknown correlations; the turbulent and pressure diffusions (Dk) and the dissipation rates (ε k) which need to be modeled.Where: Dk=¿

Finally, we end up with the following:

The Prandtl number σk connects the diffusivity of k to the eddy viscosity. Typically a value of 1.0 is used.

Where; σ t=μt

Γ t

t is the turbulent diffusivity.The turbulent diffusivity is calculated from the turbulent viscosity, using a model constant called the turbulent Schmidt number.

∂ ( ρK )∂ t

+ div ( ρKU ) = div (−PU +2 μUEij−ρU ui ' u j ' )−2 μE ij . E ij−(−ρ ui ' u j ' . Eij )

( I ) (II ) ( III ) ( IV ) (V ) (VI ) (VII )

∂ ( ρk )∂ t

+div( ρkU )= div (−p ' u '+2 μu ' e ij '− ρ 1

2u i ' . ui ' u j ' )−2 μe ij ' . eij '+(−ρu i ' u j ' . Eij )

( I ) ( II ) ( III ) ( IV ) (V ) (VI ) (VII )

Two Equation Models – The k-ε modelIn the two equation models, we develop two PDEs: one for the turbulent kinetic energy and one for the turbulent dissipation rate. The PDE for the turbulent kinetic energy is already given by the PDE for Turbulent kinetic energy, however, the expression for the turbulent or eddy viscosity is different. So, the idea is to express the turbulent viscosity as a function of K and ε and then derive PDEs for K and ε.

The equation for the model equation for k is commonly used is repeated here for convenience

The equations look quite similar; however, the k equation mainly contains primed quantities, indicating that changes in k are mainly governed by turbulent interactions.Furthermore, term (VII) is equal in both equations. But it is actually negative in themean kinetic energyequation (destruction) and positive in the turbulent kinetic energy k equation (energy transfers from the mean flow to the turbulence).The viscous dissipation term (VI) in the k equation describes the dissipation of k because of the work done by the smallest eddies against the viscous stresses.So the rate of dissipation per unit mass ε as:Now instead of modeling ε, we shall develop an independent PDE for its transport. Model equation for ε is derived by multiplying the k equation by (ε/k) and introducing model constants.The following (simplified) model equation for ε is commonly used.

The Prandtl number σε connects the diffusivity of ε to the eddy viscosity. Typically a value of 1.30 is used.Typically values for the model constants C1ε and C2ε of 1.44 and 1.92 are used.To compute the Reynolds stresses with the k- εmodel an extended Boussinesq relationship is

used The turbulent viscosity is calculated from:The Reynolds stresses are then calculated as follows:

−2 μ e ij ' .e ij '

ε = 2 ν eij ' . e ij '

Where ;(2/3)ρkδij term ensures that the normal stresses sum to k.Thek-ε model leads to all normal stresses being equal, which is usually inaccurate.k-ε model assessment Advantages:

– Relatively simple to implement.– Leads to stable calculations that converge relatively easily.– Reasonable predictions for many flows.

Disadvantages:– Poor predictions for:swirling and rotating flows, flows with strong separation,

axisymmetric jets.– Valid only for fully turbulent flows.– Simplistic ε equation.

There is many attempts have been made to develop two-equation models that improve on the standard k-ε model such as :

k-ε RNG model. k-ε realizable model. k-ω model.

The k-ε RNG(Renormalization Group Method) modelk-equations are derived from the application of a rigorous statistical technique to the instantaneous Navier-Stokes equations.Similar in form to the standard k- equations but includes:

- Additional term in equation for interaction between turbulence dissipation and mean shear.

- The effect of swirl on turbulence.- Analytical formula for turbulent Prandtl number.- Differential formula for effective viscosity.

RNG k -ε equations RNG k-ε Equations written for steady, incompressible flow without body forces.Turbulent kinetic energy:

Dissipation rate:

realizable k-εIt Shares the same turbulent kinetic energy equation as the standard k- model and improved equation for ε. The Variable Cμ is located instead of constant.

• Distinctions from standard k- model:– Alternative formulation for turbulent viscosity:

Where (A0, As, and U* are functions of velocity gradients).New transport equation for dissipation rate, :

k -ω model the basic idea of this model was originated by kolmogorov 1942 with turbulance associated with vorticity ω.In this model ω is an inverse time scale that is associated with the turbulence.and this model solves two additional PDEs:

– A modified version of the k equation used in the k-ε model.– A transport equation for ω.

The turbulent viscosity is then calculated as follows:It suffers from some of the same drawbacks, such as the assumption that μt is isotropic.SECOND ORDER MODELSThe central concept of second order models is to make direct use of the governing equations for the second order moments (Reynolds stresses and turbulent fluxes) instead of the questionable Boussinesq hypothesis. The motivation is to overcome the limitations of first order models in dealing with the isotropy of turbulence and the extra strains. The overshoot of this approach is the large number of PDEs induced which involve many unknown or impossible to find correlations. The most famous models are the Algebraic Stress Model (ASM) and the Reynolds Stress Model (RSM). THE ALGEBRAIC STRESS MODELIn the algebraic stress model, two main approaches can be undertaken. In the first, the transport of the turbulent stresses is assumed proportional to the turbulent kinetic energy; while in the second, convective and diffusive effects are assumed to be negligible. Algebraic Stress models can only be used where convective and diffusive fluxes are negligibleThis model was used in the 1980s and early 1990s. Research continues but this model is rarely used in industry anymore now that most commercial CFD codes have full RSM implementations available.THE STANDARD REYNOLDS STRESS MODEL (RSM)

The most complex classical turbulence model is the Reynolds stress equation model (RSM), also called the second-order or second-moment closure model which closes the Reynolds-Averaged Navier-Stokes equations by solving additional transport equations for the six independent Reynolds stresses.Transport equations derived by Reynolds averaging are the product of the momentum equations with a fluctuating property.Closure also requires one equation for turbulent dissipation.Isotropic eddy viscosity assumption is avoided.

• Resulting equations contain terms that need to be modeled.• RSM is good for accurately predicting complex flows.

It accounts for streamline curvature, swirl, rotation and high strain rates.The exact equation for the transport of Rı j takes the following form:

And This equation can be read as:- rate of change of Rı j=−ρ uiu j , plus- transport of Rij by convection, equals- rate of production Pij, plus- transport by diffusion Dij, minus- rate of dissipation εij, plus- transport due to turbulent pressure-strain interactions πij, plus- transport due to rotation Ωij.

This equation describes six partial differential equations, one for the transport of each of the six independent Reynolds stresses.The various terms are modeled as follows:

- Production Pij is retained in its exact form, where ;

Diffusive transport Dij is modeled by the assumption that the rate oftransport of Reynolds stresses by diffusion is proportional to the gradients of Reynolds stresses.

The dissipation εij, is related to ε as calculated from the standard ε equation, although more advanced ε models are available also.

Pressure strain interactions πij, are very important. These include pressure fluctuations due to eddies interacting with each other, and due to interactions between eddies and regions of the flow with a different mean velocity. The overall effect is to make the normal stresses more isotropic and to decrease shear stresses. It does not change the total turbulent kinetic energy. This is a difficultto model term.

Transport due to rotation Ωij is retained in its exact form.