Feasible Domain of Walker's Unsteady Wall-layer Model for the Velocity Profile in Turbulent Flows Non-profit Academic Project, Developed under the Open Acces Initiative Feasible Domain of Walker's Unsteady Wall-layer Model for the Velocity Profile in Turbulent Flows

The present work studies, in detail, the unsteady wall-layer model of Walker et al. (1989, AIAA J., 27, 140 – 149) for the velocity profile in turbulent flows. Two new terms are included in the transcendental non-linear system of equations that is used to determine the three main model parameters. The mathematical and physical feasible domains of the model are determined as a function of the non-dimensional pressure gradient parameter (p +). An explicit parameterization is presented for the average period between bursts (T + B), the origin of time (t + 0) and the integration constant of the time dependent equation (A 0) in terms of p +. In the present procedure, all working systems of differential equations are transformed, resulting in a very fast computational procedure that can be used to develop real-time flow simulators.


INTRODUCTION
Over the last forty years much effort has been placed on understanding the dynamical processes through which turbulence is created and maintained in boundary layers.The implications are evident.Provided a clear picture of the turbulence structure is developed, the basis for the construction of statistical-structural turbulence models is immediately laid down.
Wall-layer models for the innermost portions of the boundary layer are of particular interest.The extreme thinness of the viscous sublayer naturally demands the use of exceptionally fine meshes in the numerical computation of flows.To overcome this difficulty, an elegant method resides on the specification of local analytical solutions that can then be used to represent the properties of the flow throughout the wall layer.This type of approach was originally described in Patankar and Spalding (1967) and is normally referred to as the wall function method.
Most of the research on wall models is historically related to Reynolds averaged Navier-Stokes (RANS) methods; however, this approach has also appropriately served large eddy simulations (LES) of turbulent flows.Piomelli and Ballaras (2002) have reviewed the applicability of some available methodologies to UNSTEADY WALL-LAYER FLOWS observed over a large characteristic time, the quiescent period (Kline et al. 1967).To determine the mean velocity profile in the wall layer, a time-average of the leading order instantaneous velocity is performed over the average period between bursts, which is considered to be approximately equal to the duration of the quiescent period.
The present work studies, in great detail, the feasible domain of the model proposed by Walker et al. (1989).The definitions of the instantaneous and mean velocity profiles, as preconceived by Walker et al. (1989), depend on the determination of four unknowns, the average period between bursts (T + B ), the origin of time (t + 0 ), a constant of integration of the time dependent equation (A 0 ) and the local pressure gradient (p + ).Once p + is specified, a set of three non-linear equations must be solved to reveal T + B , t + 0 and A 0 .These parameters must be real and satisfy T + B > t + 0 .In Walker et al. (1989) no comments are made regarding any possible limitation on the value of p + .Our attempts to find a feasible domain p + min ≤ p + ≤ p + max from the expressions shown in Walker et al. (1989) failed.
Here, all expressions introduced in Walker et al. (1989) are verified through the Mathematica TM (Wolfram 2008) software system.In fact, it was later discovered that two terms are missing in equation ( 63) of Walker et al. (1989).The resulting non-linear set of equations for parameters T + B , t + 0 and A 0 is thus correctly presented and the feasible domain p + min ≤ p + ≤ p + max is determined.To find the numerical solution, the system of non-linear algebraic equations was transformed onto a system of ordinary differential equations with initial conditions.The system was then solved by NDSolve to generate a solution in terms of interpolation functions.
As it turns out, the model developed by Walker et al. (1989) is mathematically feasible in the domain p + 2 [-0.025, 41.886].To find this interval, the model was only required to provide real values for the parameters computed from the governing system of non-linear equations and to satisfy the condition T + B > t + 0 ; no further stringency to physical validity was required.However, if the model is required to furnish only positive derivatives at the wall for the instantaneous velocity, the feasible domain is reduced to p + 2 [-0.025, 0.104996].
The present work derives in detail all similarity solutions for the homogeneous diffusion differential partial equation presented in Walker et al. (1989).In doing so, a new treatment is introduced whereby the pressure term is included as a non-homogeneous contribution.To permit fast computations, interpolation functions were generated from initial and boundary value problems, to represent some complex special functions, including Ξ.The special function Ξ (Walker et al. 1989) and its derivatives are given exact expressions (see Mikhailov and Silva Freire 2012), based on original identities for the hypergeometric functions 1 F 1 and p F p .
The analysis of Walker et al. (1989) is specially developed for attached flow.The dominance of the error and logarithmic functions over the solution must clearly prevent its use in regions of separated flows since solutions of the type y 2 and y 1/2 cannot occur as predicted by Goldstein (1930Goldstein ( , 1948) ) and Stratford (1959).This aspect of their analysis is further discussed here.
For the first time, an explicit parameterization is presented for T + B , t + 0 and A 0 in terms of p + .These expressions make abundantly clear that p + and T + B cannot be independently specified for computations of the instantaneous velocity profile.They show that once the near wall flow dynamics is accepted to be driven by dominating diffusion effects, T + B is determined uniquely by p + .

MODEL FORMULATION
Some relevant features of the model proposed by Walker et al. (1989) are briefly reviewed to set the necessary background for discussion.Only the main aspects of the theory are presented.The properties of turbulent flows are known to experiment a complex behavior in the near wall region, with very steep changes in MIKHAIL D. MIKHAILOV and ATILA P. SILVA FREIRE mean-velocity profiles and higher-order statistics.Early studies (Prandtl (1925), von Karman (1930), Coles (1956)) of the attached turbulent boundary layer have successfully split the boundary layer into two typical regions, a viscous (inner) sublayer where turbulent and laminar stresses are of comparable magnitude and a defect (outer) layer where the turbulent stresses provoke a small perturbation to the inertia dominated external flow solution.
The identification of the pertinent length scale δ + (= v/u τ , v = kinematic viscosity, u τ = friction velocity) for the wall region permitted authors to develop local analysis, that naturally lead to analytical solutions (Prandtl (1925), Millikan (1939)) and the advance of proper dimensional arguments.For example, authors have identified the peak production of turbulent kinetic energy to occur at wall distances of the order of 12δ + (Laufer (1954), Gad-el-Hak andBandyopadhyay (1994)).

TIME-MEAN STRUCTURE
The time-mean structure of the flow in Walker et al. ( 1989) is based on the classical two-layered asymptotic analyses (Yajnik (1970), Bush and Fendell (1972), Mellor (1972)) of large Reynolds number turbulent boundary layer flow.Solutions are then developed in terms of two small parameters, R -1 and u * , where R denotes the Reynolds number based on representative external flow scales and u * = u τ / u e , u e = mainstream velocity.
Because the analysis is restricted to the inner layer, the local variables are scaled with u τ and v (kinematic viscosity).
The leading-order governing equation of the mean wall flow is set to be (Walker et al. (1989); see also: Loureiro andSilva Freire (2011), Sychev andSychev (1987), Cruz and Silva Freire (1998)), with τ and the pressure gradient parameter p + is defined through where p e denotes the external flow pressure.
The salient aspect of Eq. ( 2) is that it becomes undetermined at a point of flow separation, x s , since u τ = 0. Also note that, at this point, pressure changes greatly across the boundary layer, implying that p e is not an appropriate reference parameter at the wall (Stratford (1959), Loureiro andSilva Freire (2009, 2011), Loureiro et al. (2008)).
In the outer limit of the wall region, the mean velocity profile, U + , is required to follow a logarithmic behavior, implying that the dominant effect in Eq. ( 1) is the Reynolds stress effect.
In the outer region of the wall layer, U + must satisfy with ϰ = 0.4 and C i = 5.0.UNSTEADY WALL-LAYER FLOWS

TIME-INSTANTANEOUS STRUCTURE
To obtain the governing equations of the unsteady flow in the wall layer during a quiescent period, Walker et al. (1989) introduced the following non-dimensional variables, where L x is a characteristic length in the x-direction associated with the longitudinal extent of the outerregion structures that drive the wall-layer dynamics.
In accordance with experiments, Walker et al. (1989) consider and pick a time scale determined from the condition that the unsteady term is balanced by the viscous term in the N.-S.equations (see, Eq. ( 5)).Substitution of Eqs. ( 4) through ( 6) into the N.-S.equations together with an appropriate expansion for the pressure distribution in the wall layer and collection of the terms of leading order, furnishes the approximate governing equations.
Hypothesis (6) implies that the x-momentum equation develops independently from the other equations of motion.The flow evolution in a cross-flow plane (y + , z + ) is determined from conditions that are representative of the motions during a typical quiescent state.Solutions are then considered to be given in terms of the periodic flow development between a pair of streaks, so that the cross-flow velocity field (v + , w + ) is represented by Fourier series; appropriate wall and outer conditions are specified to reproduce the wall-layer structure.The longitudinal velocity u + is determined from conditions imposed by the outer flow and effects of the evolving flow in the cross-plane.
Solution for u + is also written as a Fourier series with coefficients u n ; substitution of the Fourier expressions for u + , v + and w + into the approximate equations of motion, yields with n = 1,2,3,..., j = | m − n |, ¸+ = non-dimensional mean streak distance and where the f n 's are the functional coefficients of the Fourier series used to describe v + and w + .
The leading-order velocity solution, u 0 , is to be found from MIKHAILOV and ATILA P. SILVA FREIRE Equation ( 8) is shown in Walker et al. 1989 (as Eq. 27) with an obvious typographical mistake.The term @ u 0 / @t + was mistyped as @ u 0 / @ y + .In fact, the time dependency on Walker et al.'s model is only accounted for by the term @ u 0 / @t + .
The set of Eqs. ( 7) to ( 9) is a coupled system of non-linear equations that has to be solved numerically.The forcing function M depends on a pressure term that must be determined from the time-dependent motions in the outer layer and on further terms arising from the evolution of the other modes.Walker et al. (1989) remark that numerical computations for large R ¸ (Reynolds number based on the mean streak spacing, ¸) show the coupling between Eqs. ( 7) and ( 8) to be weak, so that contributions to the solution of Eq. ( 8) from Eq. ( 9) may be neglected.Since this term is not considered in Walker et al.'s solution, it will not be further considered here.

SIMILARITY SOLUTIONS
Clearly, signicant contributions to the mean-velocity profile during the quiescent time are due to u 0 , which is now solely determined from Eq. ( 8).The implication is that Walker et al.'s theory of wall-turbulence can be expressed in terms of a one-dimensional diffusion equation with a source term.
The solution presented in Walker et al. (1989) considers first the homogeneous time-dependent heat transfer equation.Classical similarity methods for the homogeneous heat conduction equation consider one similarity variable and initial conditions.Equation ( 8) is non-homogeneous and is subject to boundary conditions.To extend the semi-similarity solution developed in Walker et al. (1989) to the non-homogeneous case, a new term is considered here, p + τF(η) that is (11) η = y + / 2 τ 1/2 , τ = t + + t 0 where t + 0 represents the origin of time.Substitution of Eq. (10) into Eq.( 8) yields The collection of the terms dependent on p + furnishes a differential equation for F, F'' (η) + 2 ηF' (η) − 4F (η) + 4 = 0 (13) The terms that are independent of p + furnish In classical similarity methods, separable solutions are easily obtained.The case of Eq. ( 14) is more complicated since two separation constants are required.Divide both sides of Eq. ( 14) by g(η) and use a separation constant, a, to get Next, divide Eq. ( 16) by h(τ)g(η) and use a separation constant 2α to obtain An analysis of the role of α on the problem solution is presented inWalker et al. (1989).Only solutions of Eqs. ( 17) and ( 18) for α = 0 are presented.The separation constant a is set equal to a 0 and is related to the asymptotic behavior of the time-mean profile for large η.
The solution of Eqs. ( 13) with conditions F(0) = 0 and F(∞) → 1, is given by The above solution is shown in Walker et al. (1989) with a different representation (Eq.48); however, both forms have been verified and were found to be exactly the same.
Equation ( 17) is solved with conditions g(0) = 0 and g(∞) → 1, to give To solve Eq. ( 18), we make a = a 0 and integrate directly to find where A 0 is the constant of integration.The solution of Eq. ( 15) is obtained with conditions G(0) = 0 and G'(0) = 0, and is expressed in terms of a special function, Ξ (η), that behaves logarithmically for large values of the argument.This function has been extensively studied in Mikhailov and Silva Freire (2012) and for this reason is not further discussed here.We may then write, The preceding solutions are substituted into Eq.( 10) to give the instantaneous velocity profile, This expression depends on four unknown parameters − a 0 , A 0 , t + 0 and T + B − which must be specified for prescribed pressure gradients, p + .Walker et al. (1989) proposed to determine these parameters by computing the time-average of u 0 and forcing the asymptotic form of the resulting expression in the limit of high y + to follow a logarithmic behavior.

MEAN VELOCITY PROFILE
The time-mean averaged profile (U + ) is evaluated by an integration of Eq. ( 23) over the average time between burts, T + B .Walker et al. (1989) show that this can be made analytically.The resulting expression MIKHAIL D. MIKHAILOV and ATILA P. SILVA FREIRE is very long and for this reason is not repeated here.All expressions used here in the definition of U + were checked against Eqs.( 53) through (59) of Walker et al. (1989); all results coincided.The mean velocity profile is required to satisfy conditions and the previous condition specified by Eq. (3).Condition (3) immediately gives The first condition in Eq. ( 24) gives The second condition is satisfied identically.The third condition gives (28) Equations ( 26), ( 27) and ( 28) can be solved to yield A 0 , t + 0 and T + B .They specify a system of transcendental non-linear algebraic equations that needs to be solved numerically.
In Walker et al. (1989) two terms were missing in their Eq.( 63), they are The solution of a system of algebraic non-linear equations is normally carried out in the software Mathematica TM through FindRoot.Here, the system of Eqs. ( 26) to (28) was transformed onto a system of differential equations with initial conditions, and solved through NDSolve.The special features of NDSolve resulted in a very fast computational procedure and in a very convenient solution expressed in terms of interpolation functions.This particular aspect of the present work will be discussed in detail elsewhere.
For the computations, the parameter C i in Eq. ( 26) was set constant and equal to 5. For flows under a variable longitudinal pressure gradient this certainly is not true.However, Walker et al. (1989) performed their computations with this restrictive assumption (C i = 5).So that the present results can be compared with those of the original reference, the same hypothesis was adopted here.In any case, provided C i is parameterized in terms of p + , the system of Eqs. ( 26) to ( 28) can be easily solved to determine new values of A 0 , t + 0 and T + B .See, e.g., the parameterizations presented in Mellor (1966) and Nickels (2004).In the instantaneous motion equations, scale velocities were introduced in terms of the friction velocity, Eq. ( 4).As defined in Walker et al. (1989), u τ is obtained from the time-mean structure, which is never allowed to admit a y 1/2 behavior for the mean-velocity profile irrespective of the value of p + ; as a corollary, u τ is also never admitted to be negative or zero.However, in an unsteady flow computation, if at two distinct instants of time the instantaneous velocity derivatives at the wall change sign, there must be a third instant where it is identical to zero.At this instant, the wall scaling variables need to be expressed in terms of the local pressure gradient at the wall.In fact, close to a separation point the relevant velocity scale in the wall region is u pv (= (v / p) (@ p w / @ x) 1/3 , p w = wall pressure).The conclusion is that in a same cycle, positive and negative velocity derivatives at the wall must not be allowed to occur.
To determine the pressure gradient value where flow separation is first observed, consider the extreme situation, t + = T + B , that is, the end of the cycles.Figure (5) shows that du 0 / dy + = 0 at p + = 0.104996.Therefore, if as a further requirement, the model of Walker et al. (1989) is asked to furnish only positive derivatives at the wall for the instantaneous velocity, the feasible domain is reduced to p + 2 [-0.025, 0.104996].The behavior of parameters A 0 , t + 0 and T + B can then be better observed in Figs. 6 and 7.The model is formulated in terms of some very general considerations on the observed coherent motions in the wall region.However, after many simplifications, the flow features are expected to be represented by a nonhomogeneous time-dependent, one-dimensional, diffusion equation.Effects due to the structure of the organized motions are then restricted to the specification of the duration of a cycle and to the prescription of the external pressure gradient.Despite a claim from the original authors, the model is not appropriate to describe transient reverse flows.
To develop the solutions, a special numerical procedure was implemented.The computation of the special function Ξ is particularly time consuming, therefore a special evaluation scheme was proposed 2133 UNSTEADY WALL-LAYER FLOWS (Mikhailov and Silva Freire 2012).Real time simulators for the velocity profile were implemented and can be obtained from either authors.

Figure 5 -Figure 6 -
Figure 5 -First derivative of Eq. (23) at the wall and at t + = T + B as a function of p + .

Figure ( 7
Figure (7) shows that if a fourth condition is considered, that is, ord( t + 0 / T + B ) = 10 3 , the feasible domain is reduced further to p + 2 [-0.005, 0.104996].Of course, this is an arbitrary condition based on experimental information.However, it does illustrate how Fig. (7) can be used to determine a domain of validity to the model of Walker et al. (1989) that is physically meaningful.