A NEW APPROACH TO TREAT THE RANS-LES

Transcription

1 ETMM: th International ERCOFTAC Smposium on Engineering, Marbella, 7-9 September 4 A NEW APPROACH TO TREAT THE RANS-LES INTERFACE IN PANS Lars Davidson Div. of Fluid Dnamics, Dept. of Applied Mechanics Chalmers Universit of Technolog, SE-4 96 Gothenburg lada@chalmers.se Abstract The partiall averaged Navier-Stokes (PANS) model, proposed b [], can be used to stimulate turbulent flows either as a RANS, LES or DNS. In [], a method was proposed to include the effect of the gradient off k. This approach is used at RANS- LES interface in the present stud. Three different interface models are evaluated in full developed channel flow and embedded LES of channel flow. The importance of the location of the interface in full developed channel flow is also investigated. It is found that the location and the choice of the treatment at the interface ma be critical at low Renolds number. The reason is that the turbulent viscosit, in outer scaling, is larger and hence the relative strength of the resolved turbulence is smaller at low Renolds numbers. In RANS, the turbulent viscosit and hence the modeled Renolds shear stress is onl weakl dependent on Renolds number. It is found that in the present work that also applies in the URANS region. Introduction Wall-bounded Large Edd Simulation (LES) is affordable onl at low Renolds number. At high Renolds number, the LES must be combined with a URANS treatment of the near-wall flow region. There are different methods for bridging this problem such as Detached Edd Simulation (DES), hbrid LES/RANS and Scale-Adapted Simulations (SAS); for a review, see [3]. The two first classes of models take the SGS length scale from the cell size whereas the last (SAS) involves the von Kármán lengthscale. In the present work, the PANS model is used as a zonal hbrid LES/RANS model to simulate wallbounded flow at high Renolds number. f k = in the near-wall region, and f k =.4 in the LES region. The interface between the RANS and LES regions is defined along a grid line, an approach also chosen in ZDES [4]. As an alternative to using a constant f k in the LES region,f k can be computed using the cell size and the integral length scale, k 3/ tot /ε [5], wherek tot is the sum of the resolved and the modeled turbulent kinetic energ, i.e. k tot = k res +k. The reason whf k is not computed is that it has been shown that a constant f k =.4 works well in the LES region [6]; it was even shown in [6] that using constant f k =.4 works better than computingf k, not to mention that it is numericall more stable. It was furthermore shown that the k and ε equations in the LES region are both in local equilibrium (the time-averaged equations, not instantaneousl, see Section ). In [], a method was proposed to include the effect of the gradient of f k. This ma be important, for eample, at RANS-LES interfaces. This approach is further developed in the present stud. It is evaluated in full developed channel flow and embedded LES of channel flow. in all cases, PANS is used as a zonal model. In full developed channel flow, the RANS- LES interface is parallel to the wall and in embedded LES it is parallel to the inlet. When the RANS-LES interface is parallel to the wall, the term including the gradient of f k across the interface can be both positive and negative (becausev is both positive and negative). This term is included onl when it reduces the modeled turbulence. Hence, the term can be seen as a forcing term which represents backscatter. It is used to stimulate growth of the resolved turbulence in the LES region adjacent to the RANS region. The present method reduces the gra area problem described in [7]. The paper is organized as follows. The net section describes the equations and the interface models. The following section describes the numerical method, followed b a section which presents and discusses the results. The final section presents the conclusions. The PANS Model The low-renolds number partiall averaged Navier-Stokes (LRN PANS) turbulence model reads [8] Dk = j Dε = j [( ν + ν t σ ku [( ν + ν t σ εu ) k ) ε j j ] +P k +P ktr ε () ] ε ε +C ε P k k C ε k where D/ = / t + v j / j denotes the material derivative. The constants and damping functions are given in [8]. The term P ktr is an additional term which is non-zero in the interface region because

2 /. The function f ε, the ratio of the modeled to the total dissipation, is set to one since the turbulent Renolds number is high. f k is set to in the RANS region and to.4 in the LES region. Attempts were made to compute f k in the LES region, but because of the large gradient of f k across the interface the simulations diverged. Furthermore, it has been found that in zonal RANS-LES f k =.4 in the LES region is a suitable choice [6]. It was shown in [6] that the k and ε equations in the LES region are in local equilibrium, i.e. P k ε = () ε C ε k P k Cε ε = (3) k It seems that Eqs. and 3 cannot both be satisfied since C ε Cε ; indeed, the equations are not satisfied instantaneousl for which the convective terms pla an important role. Equation 3 is satisfied but ε C ε k P k Cε ε. (4) k Equations and 3 are both satisfied because the correlation between ε, k and P k (left side of Eq. 3) is stronger than that between ε and k (right side). The correlation between the quantities on the left side is larger than that on the right side becausec ε > C ε. The Interface Condition The commutation error in PANS was recentl addressed in []. In PANS, the equation for the modeled turbulent kinetic energ, k, is derived b multipling the k tot equation (k tot = k res + k) b f k. The convective term in thek equation with constantf k is then obtained as where f k Dk tot = D(f kk tot ) = Dk (5) f k = k k tot. (6) Now, if f k varies in space, we get instead f k Dk tot = D(f kk tot ) = Dk k tot k tot (7) The second term on the right side (ecluding the minus sign) represents energ transfer from resolved to modeled turbulence. It can be written (on the right side of thek equation) k tot = (k +k res) = k + v i v i (8) LES,f k =.4 URANS, f k = wall Figure : Full developed channel flow. The URANS and the LES regions. The second term on the right side can be represented b the source term Si =.5 v i (9) in the momentum equation. Consider the momentum equation for the fluctuating velocit, v i. Multipling Si b v i = v i v i and time-averaging gives the source term in thek res equation as ( v i v i ) v i = v i v i () (. denotes time averaging). Equation is equal to the time average of the second term on the right side of Eq. 8 as it should. The signs in Eq. and the right side of Eq. 8 are different because the former is part of the k res equation whereas the latter is part of the k equation. Interface RANS f k = LES,f k = Figure : Embedded LES of channel flow. The first term on the right side of Eq. 8 can be rewritten as k = k v i v i v m v m () A corresponding term can be added as a source term in the momentum equation as Si = k v i v m v m MultiplingSi b v i and time averaging gives k v i v i v m v m k () (3) where we assume that the correlation between k and v i is weak. Equation 3 is equal to (minus) the time int

3 average of the first term on the right side of Eq. 8 as it should. In [], the second term on the right side of Eq. 7 is represented b introducing an additional turbulent viscosit, ν tr, in a diffusion term in the momentum equation as where j (ν tr s ij ), s ij = ( vi + v ) j j i ν tr = P k tr s, P k tr = ν tr s ij s ij = ν tr s = k tot (4) (5) where the right side corresponds to the left side of Eq. 8. P ktr is an additional production term in the k equation, see Eq.. In [], P ktr is re-written using Eq. 6 as P ktr = k f k (6) Modeling the Interface The gradient of f k across a RANS-LES interface gives rise to an additional term in the momentum equations and the k equation. In the present work, these terms are included onl when the flow goes from a RANS region to an LES region. The effect of these terms will reduce k and act as a forcing term in the momentum equations. Four different models are investigated. Interface Model. This is based on the approach suggested in []. The additional turbulent viscosit, ν tr, gives an additional production term, P ktr, in the k equation, see Eq. 6. Since we are interested in stimulating resolved turbulence in the LES region adjacent to the RANS region, onl negative values of ν tr are included (i.e. / < ; this corresponds to a fluid particle in a RANS region passing the interface into an adjacent LES region). However, ν tr takes such large (negative) values that ν t + ν tr <. To stabilize the simulations, it was found necessar to introduce a limit ν t + ν tr > in the diffusion term in the momentum equation. No such limit is used in the k equation, and hencep k +P ktr is allowed to go negative. Interface Model. This model is identical to Model ecept that Eq. 5 is used instead of Eq. 6. k tot is defined as k tot = k + v i v i r.a (7) where subscript r.a. denotes running average. Since k tot is mostl larger than k/f k [6], this approach will give a larger magnitude of the (negative) production. It will be seen that this modification is of utmost importance. Interface Model 3. The right side of Eq. 8 is added to the k equation, but onl when the flow goes from a RANS region to an LES region (i.e. / < ) ( ) Dfk P ktr = k tot min, (8) where k tot is computed as in Eq. 7. The production term P ktr < which means that it reduces k as it should. The sum of the Si and S i terms (see Eqs. 9 and ) in the momentum equations read ( )( ) Dfk S i = min, k.5+ v m v m v i r.a. (9) Since / <, the source S i has the same sign as v i ; this means that the source enhances the resolved turbulence as it should. It is however found that the forcing becomes too strong when S i is added to the momentum equation. In the present work S i =. The differences between Models 3 and are that No eplicit modification is made in the momentum equation in Model 3 Model (and Model ) ma need regularization in case s in the denominator of ν ttr in Eq. 5. No such regularization, however, is used in the present work. Interface Model 4. This interface model was developed in [6] for horizontal interfaces. The modeled turbulent kinetic energ in the LES region adjacent to the interface is reduced b setting the usual convection and diffusion flues of k at the interface to zero. New flues are introduced in which the interface condition is set to k int = f k k RANS (f k =.4), where k RANS is the k value in the cell located in the URANS region adjacent to the interface. No modification is made for the convection and diffusion ofεacross the interface. For greater detail, see [6]. The treatment of the k equation for a vertical interface (the embedded interface in the channel flow) is eactl the same as for a horizontal interface. The difference is that an interface condition is needed also for the ε equation which reads [] ε I = C 3/4 µ l sgs = C S ε I = C 3/4 µ k 3/ I l sgs, = V /3,C S =., () where subscript I denotes inlet or interface; V is the volume of the cell. 3 Numerical Method An incompressible, finite volume code is used. The numerical procedure is based on an implicit, fractional step technique with a multigrid pressure Poisson solver and a non-staggered grid arrangement. For

4 Re τ + z + N f Table : Grids. Full developed channel flow. f denotes geometric stretching. U + 3 the momentum equations, second-order central differencing is used for the full developed channel flow. For the embedded channel flow, we use the secondorder upwind van Leer scheme in the RANS region upstream of the interface and second-order central differencing in the LES region downstream of the interface. The Crank-Nicolson scheme is used in the time domain and the first-order hbrid central/upwind scheme is used in space for solving the k andεequations. 4 Results Full Developed Channel Flow The Renolds number Re τ = u τ δ/ν where δ denotes half channel height. Simulations at different Renolds numbers are made,re =,Re = 4 and Re = 8. The streamwise, wall-normal and spanwise directions are denoted b, and z, respectivel. The size of the domain is ma = 3., ma = and z ma =.6. The mesh has 3 3 cells in the z planes. A simulation with twice as large domain in the z plane ( ma = 6.4 and z ma = 3.) with cells was also made for Re τ = 4, and identical results were obtained as for the smaller domain. Details of the grids are given in Table. The baseline value for the location of the interface is 5. Figure 3 presents the velocit profiles, the turbulent viscosities, the shear stresses and the production terms in the k equation. The vertical thick dashed lines show the location of the interface. Model and 3 gives a velocit profile in good agreement with the log-law. Model does not sufficientl reduce the turbulent viscosit in the LES region and the result is that the turbulent viscosit is much too large but somewhat smaller than no interface condition. With Model, the magnitude of the additional turbulent viscosit near the interface is actuall larger than ν t, which makes their sum, ν t + ν tr, go negative. Please recall that in the momentum equations ν t + ν ttr is not allowed to go negative; it is negative onl inp ktr, see Section. As a consequence of the large, negativeν ttr,p k +P ktr < at the interface. The shear stresses in Fig. 3d show that without interface condition the flow goes stead. Note that to enhance the readabilit of the figure, the negative viscous plus modeled shear stresses, τ ν τ, are plotted. Recall that the total shear stress, τ tot, will (νt +νtr)/ν Pk +Pktr τ ν τ, u v (a) Velocit. +: U + = ln( )/ (b) Turbulent viscosit. : the location of the computational cell centers. 4 5 (c) Production term. : the location of the computational cell centers (d) Resolved (upper lines) and viscous plus modeled (lower) shear stresses. Figure 3: Full developed channel flow. Re τ = 4. : Model. : Model. : Model 3; : no interface condition. alwas irrespectivel of model obe the linear law τ tot =, τ tot = τ ν +τ u v = (ν +ν t ) v u v ()

5 3 U + (νt +νtr)/ν (a) Velocit. +: U + = ln( )/ (b) Turbulent viscosit. : the location of the computational cell centers. Figure 4: Full developed channel flow. Re τ = 8. For legend, see Fig. 3. With Model the flow is neither in RANS mode nor in LES mode, but it ends up in being in the middle, which is a most unfortunate condition for a hbrid LES-RANS model. The stresses predicted b Models and 3 are virtuall identical. Figure 4 shows the same quantities as Fig. 3, but now for Re τ = 8. Again, Models and 3 give good velocit profiles. Model and the no interface condition give rather good velocit profiles although the velocit is somewhat too high in the LES region. The reason for the larger velocit levels is seen in Fig. 4b which presents the turbulent viscosities. The no interface condition does not manage to reduceν t in the LES region; also Model is much less efficient in reducingν t than are Models and 3. Models 3 fail for Re τ = (not shown). In order to find the reason for this failure, the sensitivit to the location of the interface was investigated. Figure 5 presents the velocities, turbulent viscosities and the shear stresses using Model 4 for Re τ =, 4 and8 for different locations of the interface. The velocit profiles are well predicted for all cases epect for Re τ = 8 and interface = 5. Note that the turbulent viscosities are plotted using outer scaling (i.e. the are scaled with friction velocit, u τ, and half channel width, δ). The results from a onedimensional RANS simulation using the AKN model (i.e. PANS with f k = ) are also included. When plotted versus, the RANS turbulent viscosit is virtuall independent of Renolds number and hence all (ν +νt)/(uτδ) (ν +νt)/(uτδ) U Reτ = 8 Reτ = 4 Reτ = 8 (a) Velocit. +: U + = ln( )/ Reτ = 8 Reτ = 4 Reτ =...3. τ +τ ν (b) Viscosit Reτ = 4 Reτ = 8 Reτ = 5 Reτ = (c) Viscosit Reτ = 8 Reτ = (d) Viscous plus modeled shear stress Figure 5: Full developed channel flow. Model 4. Re τ = (lower lines), Re τ = 4 (middle lines), and Re τ = 8 (upper lines). Vertical dashed thick lines show the location of the interfaces. : interface = 5; : + interface = 5; : interface = ; : D RANS using AKN model.

6 the green dotted lines in Fig. 5c are the same. Actuall this is also the case for the PANS simulations when the interface is defined in outer scaling (i.e. in ): in Fig. 5b, the turbulent viscosit in the URANS region for {Re τ =, interface = 5}, and {Re τ = 4, interface = 5} and {Re τ = 8, interface = } (i.e. =.5) {Re τ =, interface = 5}, and {Re τ = 4, interface = } (i.e. =.5) are ver similar. As a result, this applies also for the viscous plus modeled shear stresses, see Fig. 5d. It is interesting to note that the viscosit in the RANS region in the PANS simulations is much smaller than that in the D RANS simulation. The reason is, as noted in [6], that low modeled k is transported from the LES region into the RANS region near the lower wall when v < (ε in the LES region in the two simulations are ver similar, see [6]). Now we need to address the question wh Models -3 fail at Re τ =. The interface location was moved awa from the wall to interface, but the same results were obtained (not shown). Net, the interface location was moved closer to the wall, to interface 5. Model -3 still fail to predict a good velocit profile (not shown). It was mentioned above that the reason for the poor prediction of the velocit profile atre τ = could be that the resolved shear stress is too low at the interface, and that the low level of resolved turbulence in the interface region would make the transition from RANS mode to LES mode difficult. However, this theor does not hold when we look at the resolved shear stresses at the interface at Re τ = (not shown) which are as large or larger than those at Re τ = 8 for which good velocit profiles are obtained. The reason for the poor prediction at Re τ = is that the turbulent viscosit (in outer scaling) is much larger than at the higher Renolds number, see Figure 5b,c. Because of this, Models -3 do not manage to sufficientl decrease ν t in the LES region adjacent to the interface. Embedded LES of Channel Flow The Renolds number is Re τ = u τ δ/ν = 95. The domain size is6.4.6 in the streamwise (), wall-normal () and spanwise direction (z), see Fig.. The mesh has cells. A geometrical stretching of.3 in the direction is used from the walls to the center. Anisotropic fluctuations are added to the momentum equations at the interface [, ]. The time-averaged interface fluctuations and the twopoint correlation of w are shown in Fig. 6. As can be seen, a constant amplitude is used across the channel. The reason for this choice instead of making the Renolds stresses var across the channel is that it is simpler and it was found in [] to be more efficient in stimulating the growth of resolved fluctuations. It (a) Renolds stresses. Figure 6: Embedded LES. Prescribed snthetic Renolds stresses at the vertical RANS-LES interface. : u rms/u τ ; : v rms/u τ ; : w rms/u τ ; u v /u τ. u v /u τ,in v /uτ,in (a) Velocit..5.5 (b) Resolved shear stress Figure 7: Embedded LES. Re τ = 95. = 5.9. For legend, see Fig. 3. can be noted in Fig. 6 that the sign of the shear stress changes at the centerline. This was achieved b simpl switching the sign of the vertical snthetic fluctuation, v, for >. When creating the snthetic turbulent interface fluctuations, the length scale was set to.5. Figure 6b shows that the two-point correlation of the generated fluctuations agrees well with this value. A Matlab file to generate the snthetic fluctuations can be downloaded at []. RANS is used upstream of the interface (f k = ) and LES is used downstream of it (f k =.4). Models -3 are used at the vertical interface. Model 4 was used in [] and was found to work well.

7 uτ ma (u rms (,)) ma ((ν tot(,)/ν (a) Friction velocit. : u τ, RANS for fulldeveloped channel; : u τ, PANS for fulldeveloped channel (b) Streamwise fluctuations 4 6 (c) Turbulent viscosities. ν tot = ν +ν t +ν tr. Figure 8: Embedded LES. Re τ = 95. Vertical thick dashed lines show the RANS-LES interface. For legend, see Fig. 3. Figures 7 and 8 present velocit profiles, resolved turbulence, skin friction, turbulent viscosit and production of modeled turbulent kinetic energ. First, it can be noted that without an interface model, little resolved turbulence survives in the LES region because it is killed b the large turbulent viscosit. Also Model does not reduce the turbulent viscosit sufficientl, although it is not as bad as the no interface model. For Model, some resolved turbulence survives but it is small. The friction velocit (Fig. 8a) is the most critical parameter. At the right of the figure, the full-developed values for PANS and RANS are indicated. As can be seen, Models and 3 give actuall much better value of u τ (the target value is one) than the full-developed values; for a ver long channel, however, the u τ values will approach the values indicated at the right. Models and 3 give u τ alread two half-channel heights downstream of the interface, which is quite good. The resolved turbulence is fairl good near the wall, but it will take some time (i.e. streamwise distance) to reach their fulldeveloped profiles; the resolved shear stress (Fig. 7b) which is the most important is fairl well developed. The profiles of the normal resolved stresses still remember the constant profiles imposed at the interface (not shown). But, as mentioned above, the advantage of using a constant imposed interface profiles is that it is ver efficient in creating resolved turbulence. In the case of Model, the resolved turbulence further downstream ( 6.4) will probabl first be completel killed and then the flow will eventuall go into RANS mode. With Models and 3, the resolved turbulence at the interface increases strongl (Fig. 8b) thanks to the added non-isotropic fluctuations, but also thanks to interface conditions of Models and 3 which succeed in drasticall decreasing the modeled turbulent kinetic energ and thereb the turbulent viscosit, see Fig. 8c. The turbulent viscosit is actuall completel killed for. The turbulent viscosit is reduced alread upstream of the interface because f k / < at the computational cell upstream of the interface. The turbulent viscosit with Model 3 is reduced one cell further downstream compared to Models and. The reason is that Model 3 does not modif the turbulent viscosit in the momentum equations, but it modifies onl the production, P k, b adding P ttr. Models and modif the turbulent viscosit b addingν ttr in the momentum equations (the modif also production throughν ttr ). Careful inspection of Fig. 8c shows that Model does indeed reduce ν t at the interface, but contrar to Model the turbulent viscosit increases directl downstream of it. The reason is that Model creates much larger negative ν ttr than does Model ; for Model ma( ν ttr )/ν 3 and for Model ma( ν ttr )/ν 5. For Model the maimum is located in the computational cell downstream of the interface becausek tot = k+k res (see Eq. 5) is largest here. Upstream of the interface, Model gives ma( ν ttr )/ν 9. As for the full-developed channel flow, it is found that the productions b the interface models, P ktr, are ver large and negative for Models and 3 (not shown). These large negative productions ensure the strong decrease in turbulent viscosit across the RANS-LES interface, see Fig. 8c, which in turn facilitates the rapid increase in resolved turbulence downstream of the interface, see Fig. 8b. 5 Conclusions Simulations using Zonal PANS are made where f k = in the URANS regions and f k =.4 in the LES regions. There is a a strong gradient of f k across the RANS-LES interface. Recentl a modification of the PANS model was proposed b [] where the take the gradient of f k into account. This approach is used at the RANS-LES interfaces and it is further devel-

8 oped and evaluated in the present work. Four different treatments of the interface are evaluated, one is the approach proposed in [] (called Model ), Models and 3 are modifications of Model and Model 4 is the approach presented in [6]. The beaut of Models -3 is that the do not depend on an constants or tuning; the are given b the gradient of f k across the interface. The disadvantage of Model 4 is that is depends on how the modeledk andεare prescribed at the interface. The four models are evaluated in full developed channel flow and embedded LES in channel flow. Model does not give good results. Its effect is too weak and the model does not sufficientl decrease the turbulent viscosit on the LES side of the interface. Models and 3 give mostl good results and alwas virtuall identical results. To make Models and numericall stable, the additional turbulent viscosit, ν ttr, from the models must be positive in the momentum equations. In the production term in the k equationν ttr is permitted to go negative. No modifications are made in the momentum equations in Model 3. The magnitude of the negative added production, P ktr, in the modeled k equation is ver large, much larger than the ordinarp k. This is the ke to the success of Model and 3;P ktr is ver large and negative in a few cells straddling the interface which reduces k and thereb the turbulent viscosit. Models and 3 fail for the full developed flow at the lowest Renolds number (Re τ = ); the both give stead RANS results. Model 4 works well for all flows. In full developed channel flow using RANS, the turbulent viscosit, and hence the modeled Renolds shear stress, is virtuall independent of the Renolds number. Using Zonal PANS, it is found that this also applies for the turbulent viscosit and the modeled Renolds shear stress in the URANS region. The conclusion is that interface Model 4 works best and that Models and 3 fail in one flow (full developed channel flow at Re τ = ). The disadvantage of Model 4 is that it depends on the prescribed boundar conditions of k and ε at the RANS-LES interface. References [] S.S. Girimaji. Partiall-Averaged Navier-Stokes model for turbulence: A Renolds-averaged Navier-Stokes to direct numerical simulation bridging method. Journal of Fluids Engineering, 73():43 4, 6. [] S. S. Girimaji and S. Wallin. Closure modeling in bridging regions of variable-resolution (VR) turbulence computations. Journal of Turbulence, 4():7 98, 3. [3] J. Fröhlich and D. von Terzi. Hbrid LES/RANS methods for the simulation of turbulent flows. Progress in Aerospace, 44(5): , 8. [4] Sbastien Deck. Recent improvements in the zonal detached edd simulation (zdes) formulation. Theoretical and Computational Fluid Dnamics, 6(6):53 55,. [5] B. Basara, S. Krajnović, S. Girimaji, and Z. Pavlović. Near-wall formulation of the Partiall Averaged Navier Stokes turbulence model. AIAA Journal, 49():67 636,. [6] L. Davidson. The PANS k ε model in a zonal hbrid RANS-LES formulation. International Journal of Heat and Fluid Flow, pages 6, 4. [7] A. Travin, M. Shur, M., Strelets, and P. Spalart. Detached-edd simulations past a circular clinder. Flow, Turbulence and Combustion, 63:93 33,. [8] J. Ma, S.-H. Peng, L. Davidson, and F. Wang. A low Renolds number variant of Partiall- Averaged Navier-Stokes model for turbulence. International Journal of Heat and Fluid Flow, 3(3):65 669,. [9] H.K. Versteegh and W. Malalasekera. An Introduction to Computational Fluid Dnamics - The Finite Volume Method. Longman Scientific & Technical, Harlow, England, 995. [] L. Davidson and S.-H. Peng. Embedded large-edd simulation using the partiall averaged Navier-Stokes model. AIAA Journal, 5(5):66 79, 3. [] L. Davidson. Using isotropic snthetic fluctuations as inlet boundar conditions for unstead simulations. Advances and Applications in Fluid Mechanics, (): 35, 7. [] L. Davidson. How to implement snthetic inlet boundar conditions. lada/projects/inletboundar-conditions/proright.html, 3.