PERTURBATION ANALYSIS OF ” k − ω ” AND ” k − ” TURBULENT MODELS . WALL FUNCTIONS

Presented article shows rigorous method how derive non-stationary turbulent boundary layer equations by perturbation analysis. The same method is used for analysing behaviour of ”k-omega” and ”k-epsilon” turbulent models. The analysis is divided into two parts: near wall behaviour boundary conditions, and behaviour in ”log-layer” wall functions. Both parts have important place in CFD. Boundary conditions are important part of CFD. ”k-omega” and ”k-epsilon” are related by one simple formula, but they yield to different solutions. Exact values for k, omega and epsilon on a wall are evaluated and all theoretical results are compared with numerical solutions. Special treatment is dedicated to ”k-epsilon” model and Dirichlet boundary condition for ”epsilon, instead of standard Neumann boundary condition. ”Log-Layer” is well known from experiments and it is used for setting constants in turbulent models. Standard equations are derived by perturbation analysis. In presented article are these equations derived with 3 more terms, than in standard case. This yields to sharper approximation. These new equations are solved and solution is a bit different, than in standard case. Due to 3 extra terms is possible to get better approximation for k and new view into problematic..


Introduction
Two equations turbulent models are widely used in CFD -they give "good" results in "good" time.The most common are "k − ω" and "k − ε".This article is about turbulent boundary layer equations and perturbation analysis of turbulent models -on a wall and in "log-layer".Main goal is deriving Mathematical formulas for "k", "ω" and "ε". on a wall -boundary conditions, and in "log-layer" -wall functions.

∂T ∂y
Where a e-mail: vostruha@vzlu.czb e-mail: pelant@vzlu.cz We need to close the Navier-Stokes equations by a turbulent model."k − ω" turbulent model: Where Frequently used term is friction velocity defined as u τ = ν ∂u ∂y w .

Turbulent boundary layer equations
We assume a smooth flat plate and a vicious, incompressible turbulent flow.We use neglecting method which is based on magnitudes of Reynolds number and distance from the wall.
Re T is function of coordinates, but we assume for purposes of neglecting method, that Re T = C √ Re .This approximation fits only in some hight up to the wall.Our goal is studying near wall behaver and we will see, that value of the constant C isn't significant.We define the neglecting parameter: We introduce new variables Asymptotic expansion of (u, v): We put new variables and asymptotic expansions into incompressible turbulent the Navier-Stokes equations in nondimensional form.We neglect all elements containing ε, then we get turbulent boundary layer equations: We should easily return to dimensional equations: Equations ( 1) -(3) were derived without respect to used turbulent model.We can use the same method to "k − ω" and "k − " turbulent models and get behaviour of these models in the turbulent boundary layer

Behaviour on a wall and boundary conditions
We assume in all cases a stationary, incompressible flow, ∂ ∂x ∂ ∂y , v u. and ν T ν.These assumptions are correct and we will see another reason to use these assumptions later.We starts with equation for "ω".
Comparing with DNS tells, that ω = O(y −α ), velocity profile on wall is linear: u(y) = y u 2 τ ν .By this way we should neglect ∂u ∂y .Then we get equation for ω, which was derived by Wilcox: We use ansatz ω = Cy −α and it yields to: Formula ( 6) is used for evaluating values of ω until the third line of cells.
We know, that turbulent kinetic energy decrease to 0 on a wall.We use previous assumptions for deriving equation for k and ansatz k = Cy α : Equation for is from formula = kω.
Suitable boundary conditions are: k = 0 or ∂k ∂n = 0.If we use = kω, we see = O(1) and there is only one suitable boundary condition ∂ ∂n = 0. Any Dirichlet boundary condition is unsuitable due to friction velocity.We will have to know values of friction velocity during all times.In case of boundary conditions is "k−ω" better then "k− " model.We can see from approximations for k and ω, that ν T = O(y 4 ).

Wall functions
Wall functions are approximations of k, ω or in so-called "log -layer".Velocity profile is described by " the law of the wall" u(y) = u τ κ (ln y + C).Where κ is von Karmar constant and constant C depends on roughness of the wall and friction velocity.In this layer 1 Re T .Then equations are: We assume an infinite plate parallel to the flow : p w = const., ∂ ∂x ∂ ∂y , "log-layer": u = u τ κ (ln y + C), then the system of equations is over-setted.It means, that we have 4 equations and 3 unknown functions.We will see, the system has point solution only on one line.
Equations ( 8) -(10) under the assumed "log-layer" approximation have solution: Equation ( 11) yields to point solution on the critical line: k on the critical line is: We use well known formula for turbulent viscosity in "loglayer" Then we can evaluate: In light of the last formulated formulas and formula for k crit we get The right one wall functions on the critical line (y = y crit ) are: With the same "log-layer" assumptions.Solution (standard wall functions): If we use lim C 2 →0+ on solutions ( 12) -( 14) and concept of critical line (16), we get "standard wall functions".But y crit → +∞ The limiting process shows, that presented theory is more complex and explain why "standard wall functions" yields to poor results.Another shape of this theory is impossibility to find extension under the same set up.
We have to figure out by experiment value of constant C 1 , we don't need to know value of constant C 3 , but we assume it's dependence on u τ .Value of y crit has to evaluated from empirical formula.Following chapter shows limits Of "simple" methods in boundary layer theory.

Impossibility theorem
We saw in previous section approximation of vertical part of velocity by a constant.In this section we formulate and prove theorem, that we can't use any method of self-similar solutions, which are well known from laminar boundary layer theory Theorem 1 Self-similar solutions can occur only in laminar boundary layer.
Proof We create proof by contradiction.Let exist a selfsimilar solution of an incompressible turbulent boundary layer equations described by this system of equations with a turbulent model.
With a turbulent model, we don't need equations from the turbulent model.We have these boundary conditions: We assume a self-similar solution, we need to find proper boundary conditions.. We can find parts of velocity in this form: We put formulas ( 19) and (20) into the equation ( 18) and we seek for self-similar solutions.It yields to this condition: Solution is: Constant C ≤ 0, or we get infinite viscosity.It yields to this system of inequalities: We get contradiction -ν T = 0 everywhere in turbulent boundary layer.We can see sharp border between methods of laminar and turbulent boundary layer.

Comparison and figures
We compare theoretical results with numerical solutions of compressible flow in this section.Main goal is check out how incompressible flow is good approximation for compressible flow.Numerical solution was done with k − ω model with these constants: α = 5 9 , β = 3 40 , β * = 9 100 , σ = σ * = 1 2 .Computation mesh: 245 x 64.Outer velocity 30m/s.The first level of knots of the mesh 2.10 −6 , quotient if increasing high q = 1.2.Coordinate axes are in logaritmic scaling fpor better view 01097-p.4We can see velocity profile of u + = u u τ (blue curve) compared with "log-profile" (red curve)in figure 1.κ = 0.54, the value is much higher, than in literature.We can see one point intersection.
We can see evolution of yω in figure 2. We can see constant part.We compared the value of constant part with evaluated value for ω: We can see evolution of the turbulent kinetic energy (blue curve) and comparison with standard wall function (bared line) for k in figure 3. The comparison is good due to κ = 0.54 -constant line.Wall behaviour (7) (orange curve) fails.

Conclusion
We saw in section 4 the most complex wall functions for k − ω model based on fully developed "log-layer" and their link to "standard wall functions".These new wall functions explain problems in using standard ones and shows, that logarithmic approximation isn't the best option for k − ω model.Futher research will be focused on extending presented theory on whole log-layer by using special perturbation functions.Setting boundary conditions on the critical line is complicated for numerical solutions and preparing commutation grids.The greatest disadvantage of all wall functions based on log-layer is strong assumption in stationary flow with fully developed log-layer.Near to point of separation or in non-stationary flows are standard boundary conditions better choice.
Compartment showed poor results and hight κ.We explain it as comparison of compressible and incompressible flow, not enough dense mesh, σ * = 0.5 instead σ * = 1 as was evaluated.More tests are required and work extension of new wall functions and change them by perturbation functions and constants into useful form in compressible flow.Then wall function become power tool in modelling of atmosphere or other cases, where is problem to set boundary conditions on walls (surfaces)

Fig. 2 .
Fig. 2. yω .C 1 .C 2 are constants.k is called kinetic turbulent energy, is called dissipation of turbulent energy.ω is called specific dissipation and it was introduced by Wilcox.Relation between and ω is: * , σ, σ *