for Two-Dimensional Wake Flows Jerey S. Danowitz Saul A. Abarbanel Eli Turkel Abstract

Transcription

1 A Far-Field Non-reecting Boundary Condition for Two-Dimensional Wake Flows Jerey S. Danowitz Saul A. Abarbanel Eli Turkel August, 995 Abstract Far-eld boundary conditions for external ow problems have been developed based upon long-wave perturbations of linearized ow equations about a steady state far eld solution. The boundary improves convergence to steady state in single-grid temporal integration schemes using both regular-time-stepping and local-time-stepping. The far-eld boundary may be near the trailing edge of the body which signicantly reduces the number of grid points, and therefore the computational time, in the numerical calculation. In addition the solution produced is smoother in the far-eld than when using extrapolation conditions. The boundary condition maintains the convergence rate to steady state in schemes utilizing multigrid acceleration. School of Mathematical Sciences, Tel Aviv University This research was supported by the National Aeronautics and Space Administration under NASA Contract No. NAS-98 while the second and third authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA i

2 Introduction Introduction Numerical solution schemes for nonlinear ow equations usually require the introduction of one or more articial boundaries to be placed at some distance from a body around which (in external ows) the ow takes place. On these boundaries appropriate boundary conditions must be established that are not only physically correct, but that also comply with mathematical requirements as well. In, for example, subsonic ow regimes, it is well known that for hyperbolic ow equations (e.g. Euler Equations), one characteristic points back into the ow domain at outow, and therefore one condition must be prescribed; the rest, called numerical boundary conditions, must be in some way consistent with the partial dierential equations. This paper deals with the derivation of outow (far-eld) boundary conditions. These boundary conditions will be both temporally and spatially local. The proper formulation of far-eld conditions remains a vexing problem to date, and many authors have tried to tackle it in a variety of directions. It was once common in steady state calculations to set p = p,(pbeing pressure) and to, say, extrapolate all other quantities. Certainly this procedure is inappropriate for boundaries close to the body, and so alternate approaches are needed to be developed. At present, the Navier-Stokes solver, [?], uses extrapolation on all physical quantities in subsonic regimes (note that in supersonic regimes, extrapolation of all variables is correct from a mathematical standpoint) with great success. In fact, its authors claim that they have been unable to replace far-eld extrapolation with another far-eld condition and to achieve satisfactory steady state convergence [?]. It has become popular to place \non-reecting" boundary conditions at outow in subsonic ows. While it is true that generally one cannot develop local boundary conditions that give no reections, one can consider conditions which to some degree are better than others. The notion of better in this case implies that (see [?]) the reections are decreased and steady state convergence is accelerated. the reections will decrease as the position of the articial boundary tends to innity. as the incident wave is more normal to the articial boundary, the reections decrease. In 988 Abarbanel, Bayliss and Lustman (ABL) [?] developed a non-reecting outow boundary condition for viscous, compressible ows over a at plate. Their idea was to linearize the Navier-Stokes equations around a far-eld steady state solution Far-Field Non-reecting Boundary Condition August, 995

3 Introduction δ Inflow flow direction airfoil l wake region δ y x free stream region Figure : External ow topology for compressible viscous ows around NACA with an assumed u-velocity prole based upon an asymptotic approximation to the equations themselves. These linear perturbation equations were then assumed to take on a dispersive wave solution and the resulting set of equations (similar to the Orr- Sommerfeld system) described the asymptotic behaviour along the outow boundary { the articial boundary. Abarbanel, Bayliss and Lustman then assumed that the decay rate of the perturbations is mainly controlled by the iteration scheme's inability to dissipate long wave disturbances. So they expanded the system, including the wave frequency, around small wave numbers i.e. long waves. The resulting zeroth and rst order eigensystems are solved for their eigenvalues. The eigenvalue of the zeroth order eigensystem physically corresponds to the decay rate of the long wave disturbances, and the eigenvalue of the rst order eigensystem physically corresponds to the group velocity of these disturbances. The slowest decay rate is found, and it denes uniquely the group velocity. The details of the aforementioned strategy may be found in [?]. The outow boundary condition derived from this takes on + (p (:) where > is proportional to the decay rate of the disturbances, and < is their group velocity. This condition is very eective in the at plate case. This paper uses the ABL approachindeveloping a far-eld nonreecting boundary condition for two-dimensional wake ows. We consider two cases. The rst being compressible viscous laminar ows around NACA, as illustrated in gure??. In this case the compressible Navier-Stokes equations are numerically integrated using Far-Field Non-reecting Boundary Condition August, 995

4 Introduction 3 inow boundary free stream region outow boundary wake region at plate Figure : The at plate topology for incompressible viscous boundary layer ows a time consistent scheme. Multigrid acceleration has been used in this case as well. The second case considered is incompressible viscous boundary layer ow over a nite at plate as illustrated in gure??. In this case the incompressible boundary layer equations are numerically integrated both time consistently and with local-time stepping. The nonreecting boundary condition has been developed accordingly for timeconsistent cases with and without multigrid acceleration and for local-time-stepping cases without multigrid acceleration. In x we outline the long wave asymptotic expansion procedure in the general compressible case and develop the appropriate eigensystem for this case. In x3 we consider the incompressible boundary layer case and extend the boundary condition to local-time-stepping schemes. Then in x we derive the non-reecting boundary condition both with and without multigrid acceleration. In x5 we present numerical results beginning with the numerical procedure used to obtain the eigenvalues needed in building the boundary condition, and ending o with a general discussion of the results of all the numerical tests performed. In x6 we comment on the results. Far-Field Non-reecting Boundary Condition August, 995

5 Long Wave Asymptotic Expansions Long Wave Asymptotic Expansions The Navier-Stokes equations governing two-dimensional compressible ows may be written in the following form: 8 : continuity : t +(u) x +(v) y = x momentum : (u t + uu x + vu y )+p x =(u xx + u yy + (u 3 x + v y ) x ) y momentum : (v t + uv x + vv y )+p y =(v xy + v yy + (u 3 x + v y ) y ) energy : c v DT Dt + p(u x + v y )=r(rt)+ (:) where is the density, u and v are the x and y components of the velocity and p is the pressure. In addition,, the heat conductivity, c p and c v, the specic heats at constant pressure and volume, and their ratio, c p =c v are assumed to be constant. The viscous dissipation function, is dened as: =(u x +v y ) +(u x +v y )+(u x+v y ) (:) and we shall use the Stokes relation, + =. The equation of state for an ideal 3 gas, p T = (:3) c v c p is used. We now dene the Prandtl number, P r cp, which will be used in what follows. Also the Reynolds number with respect to the length of the airfoil is dened as R l U l (:) where the subscript, implies that the \free-stream" value of the quantity is taken. In addition, the viscosity, is assumed to be constant. Assume that the ow is external, and the length of the airfoil is l. Dening the dimensionless variables: s = x=l and z = y=l, an approximate wake prole in the zero lift case can be obtained [?]: u U = p Rl z m p s e R l s : (:5) Z + Integrating (??) we obtain an expression for m in terms of the drag coecient, c D : u m = ( ) dz = c D U : (:6) Far-Field Non-reecting Boundary Condition August, 995

6 Long Wave Asymptotic Expansions 5 8 We substitute the above into (??) and make the following parabolic approximation: u (z; s ) def U : p cd Rl h p s R l i z if 9s z < 9s R l l (:7) if z 9s R l where x = s l is far enough from the airfoil so that the value of m using (??) will not change as s>s. Note that (??) satises (??), and denes an approximate wake thickness of: = 3 s s R l : Perturbing (??) in the region y < around steady state conditions far downstream gives: 8 : + u t (y) + x u + x v = y [u +u t (y)u +(u x ) y v ]+p x =[u + xx u ]+ yy 3 [u x +v] y x [vt+u (y)vx]+p y =[v xx + vyy]+ 3 [u x+vy] y p t+p u x + p v y + u (y)p x = P [(p r xx + p yy) p ( xx + yy)]+( )(u ) y [u y + vx] (:8) where: u = u u, p = p p, = and v = v, and p = p, =, v =, u (y) is dened in (??). () =. then Let K = u (y) =U [K+( c d p Rl p s (:9) K) y ]: (:) Note that in the wake region, <K< and outside of the wake region, K =. Also, we make the following dispersive wave ansatz: 6 u v p = e i( t+bx) 6 F (y) F (y) F 3 (y) F (y) Next, we non-dimensionalize in the following manner: 3 75 : (:) F = G F = U G F 3 = U G 3 F = U G = U! b = " = d y = = d U dy d (:) Far-Field Non-reecting Boundary Condition August, 995

7 Long Wave Asymptotic Expansions 6 where = = (kinematic viscosity). Substituting the ansatz and the non-dimensionalizations into (??) leads to the following set of dimensionless perturbations (similar to the Orr-Sommerfeld system) in the wake region ( ). i[! + K(+( K K )]G + ig + G 3 = i[!+k(+( K K )]G + ( K)G 3 + ig = "[G + i 3 G 3 i[! + K(+( K K )]G 3 + G = "[ 3 G 3 + i 3 G G ] i[! + K(+( K K )]G + ig + G M 3 = M " P r [(G G M ) (G G M )] + ( )( K)"[G + ig 3] 3 G ] (:3) where () = d d. We now perturb (??) around long wavelengths { about small. To this end, we substitute: G i = G () i + G () i + G () i + (i=;;3;) (:)! =! +! +! + into (??), obtaining the following zeroth and rst order problems: 8 : The Zeroth Order Problem: g ' = g +g ( K)' = ' + 3' 3 = g ( )( K)P r M g + P r ' " M[ + Pr ]=: where g i G () i ;i= ;;3; together with the change of variables: (:5) i! = " g 3 = "' g = " : (:6) The First Order Problem: 8 : G = i g + i [! + K( + K K )]g G +G ( K) =i[! +K( + K K )]g + i" [ + 3! 3 = i 3[! + K( + K K i )]' g G ( )( K)P r M G + P r "M[ + Pr ] = i" P r M[! + K( + K K )] ip r g +" i( )( K)P r M ' 3 ' ] (:7) with the following change of variables: G = "G () G = "G () = G() 3 = G () : (:8) " Far-Field Non-reecting Boundary Condition August, 995

8 Long Wave Asymptotic Expansions 7 In order to solve the zeroth and rst order problems, appropriate boundary conditions need to be established. We shall derive these conditions by looking at the perturbations outside the wake region, K =, and then match at=. One can easily check that placing K = in (??) gives a system with constant coecients. We shall call the perturbations outside the wake region, Q, analogous to G in the wake region. Clearly this implies that Q i takes the form ^Q i e r with ^Q i as constant. Now the exact boundary h conditions h for the zeroth and rst order systems are G i () = Q i ();i =;;3;. In order to get a relationship on the G i 's we dq use the fact that they satisfy i d i= = dg i d i=. Since also Q i must satisfy h dq i d i= =[rq i] = i =;;3;, then clearly appropriate boundary conditions on G i () are: " dgi =[rg i ] d = i =;;3;: (:9) Q i = In order to obtain r in the previous expression, we do the following: Substituting K =into (??) and using Q i = ^Q i e r and hence Q i = rq i, and = r Q i we obtain: 8 : i(! + )Q + iq + rq 3 = (i(!+) "r )Q i " rq iq = (i(!+) 3r )Q 3 i " rq 3 + rq = (i(!+) P r "r )Q + " P r M r Q + i Q M + r Q M 3 =: (:) We expand Q i and r around as in (??) and dene Q () i zeroth order problem in the free stream region: 8 : i! q + r q 3 = (i! "r)q = (i! 3 "r)q 3 + r q = " P r r q +r q 3 +(i! r P r ")M q = = q i and obtain the (:) where it is evident that either q =orr = i! = : We note that e r cannot " decay as!if r = hence q =. For the rest of the system, the rst, third and fourth equations in (??), we apply the existence criterion for nontrivial solutions of homogeneous systems, namely: det B@ i! r i! 3 "r r " P r r r M(i! P r "r ) CA = (:) Far-Field Non-reecting Boundary Condition August, 995

9 Long Wave Asymptotic Expansions 8 We expand this around the rst row retaining terms up to O(" M). Assuming that, then, we nd that: 3 " M r = ( " M P r [ +( 3 P r )" M] (:3) For subsonic ows with =: and P r =:7 and small " (physically reasonable) the latter result is negative. Hence the perturbations will not decay in. Therefore, we choose: r = M " (:) where the sign is taken for positive and the + sign is taken for negative, so that the perturbations in free-stream will die out in. 8 : The rst order problem in the free stream region takes the form: i! Q () + r Q () 3 = i(! +)q r q 3 (i! "r)q () = i " r 3 q 3 iq (i! 3 "r )Q () 3 + r Q () =( i(! +)+ 8 3 "r r )q 3 r q " P r r Q() + r Q () 3 +(i! P r "r)m Q() = ( i(! +)+ P r "r r )M q "r r P r q r q 3 : (:5) The r chosen above makes the coecient determinant of the left-hand-side singular. Hence, solvability here requires that the same coecient determinant with one of its columns replaced by the right-hand-side be singular as well. This criterion leads to: r = i(! +)M +O(" M ): (:6) In order to x the boundary conditions for (??) and (??), we recall the requirement that: " dgi =[rg i ] d = i =;;3; whose expansion is d d = hg () i + G () i + i= =h (r +r + )(G () i + G () i + ) i= gives ( () G i () = r G () i G () i () = r G () i () + r G () i : (:7) This, together with q =$g () = provides us with appropriate boundary conditions for the zeroth and rst order perturbation problems. The reader will note Far-Field Non-reecting Boundary Condition August, 995

10 The Incompressible Case 9 that the systems (??) and (??) are of order six, so only six conditions will be prescribed for them. For physical reasons, we do not place conditions on the pressure perturbations, i.e on and on. In summary, the boundary conditions on (??) are: g () = r g () g ()= ' () = r '() (:8) and on (??) are: G () = r G () + "r g () G () = r G () () = r () + "r '() (:9) Since the perturbations decay like e "t,wesolve the zeroth order problem for its minimal positive and then use this in the rst order problem to nd! using a Fredholm like process. Since d! j d ==!, then! has the physical meaning of being the group velocity of the longest waves. 3 The Incompressible Case In this section we shall nd appropriate and! in the special case of incompressible viscous ows, i.e. M =. 3. The Time Consistent Case 8 : The zeroth order problem in this case is: g ' = g +g ( K)!' = ' + 3 ' 3 = g +P r ' = g ()= g ()= ' ()= whose normalized solution, in the sense that k g k L = is as follows: ( ;) (3:) g = ' = = g ()=cos( ) (3:) hence: < min = : (3:3) Far-Field Non-reecting Boundary Condition August, 995

11 The Incompressible Case The rst order problem is interesting in that looking at (??) one would think that the appropriate boundary conditions on G () are G () = making the usage of the Fredholm alternative incorrect. It can be easily shown that in fact: 8 : G = i g G +G ( K) = i[! + K( + K K )]g + 3! 3 = i g G + P r = ip r g G ()= G ()= ()= (3:) so that left-hand-null-vector of the rst order problem is indeed cos( ). Taking the scaler product of cos( ) and the second equation in (??) gives! = K ( K)( 3 6 ) : (3:5) It should be made clear at this point that had the condition g() = been used instead of g () =, in the zeroth order problem, we would have found that g () = sin( ). In this case we would not have been able to solve the rst order system for!. Soitisvery important to identify the proper boundary conditions in order to assure nding and!. The longwave expansion asymptotics done thus far are based on the assumption that the system of partial dierential equations are solved in a time consistent manner using a global time marching scheme. In addition, had we started with the viscous incompressible boundary layer equations ( ux + v y = u t +(u ) x +(uv) y = u yy (3:6) and done the same type of longwave asymptotics as in x (note that the far-eld prole, (??) is appropriate for this case as well) we would have obtained the same and! as above. This means that essentially we have found the eigenvalues for the viscous incompressible boundary layer equations for time consistent numerical schemes. In the next subsection, we shall nd these eigenvalues when local time stepping is to be used. 3. The Local Time Stepping Case Many researchers use local time stepping in numerical marching schemes towards steady state. While local time stepping is not time consistent, it has been found to accelerate convergence to steady state. We return to the viscous incompressible at-plate boundary layer equations, (??). Far-Field Non-reecting Boundary Condition August, 995

12 The Incompressible Case We are interested in integrating this system numerically. In order to ensure numerical stability, the Courant Fredrichs Levy (CFL) condition on the time step for each grid node requires (see [?]) that t be bounded by some function of x ij and y ij. The function depends upon the numerical scheme used. In our numerical experiments Runge Kutta schemes were used. We were conservative in our approach and required that t min ij (3:7) ij where ij min( xij u ij + ;(y ij) ) (3:8) where <. The addition of the factor helped stabilize the numerical iterative process. For the runs with global time stepping on a grid with constant xand y, =was enough for stability. however for the runs on a grid with nonconstant y, wechose = to achieve stability (see x5). Local time stepping means that at the node (x ij ;y ij ) a time step of ij is chosen. Using local time stepping one discretizes the temporal derivatives in the following un+ ij u n ij : ij ij Suppose we discretize (??) on a non-uniform rectangular grid and temporally discretize as in (??). Provided that the spatial derivatives are discretized ij ij =, signifying that the grid is not changing in these spatial discretizations are in fact valid approximations for the spatial derivatives. However, the temporal derivative is quite dierent. Since u n+ ij u n ij t! u n+ ij u n ij = ij t ij we clearly see that what we are approximating is in fact t lim : (3:) Suppose that the grid was constructed so that in the boundary layer, ij = (y ij) and that the grid is geometrically expanding in the y direction so that (3:) y ij = k j y i ; k Far-Field Non-reecting Boundary Condition August, 995

13 The Incompressible Case then and therefore t t! lim = t! ij ij! h +! = yi y ij k y y i (k ) i Note that <F(y;k)< when k> and that F (y;)=. def F (y; k): (3:) Therefore, while we are solving (??) on this special grid using local time stepping, we are in actuality solving the modied system: ( ux + v y = F(y;k)u t +(u ) x +(vu) y = u yy (3:3) We shall now carry out a long-wave asymptotic analysis for the above system of equations. We note that the far-eld wake prole, (??) is still appropriate in this case. Hence, using the parabolic far-eld approximation for the wake prole (??) we again dene the primed variables u = u u v = v: Perturbing (??) around steady state variables retaining linear terms in primed variables and substituting the dispersive wave ansatz: " u t+bx)" v = e i( F (y) (3:) F (y) and nondimensionalizing, b, ", and y as in (??) we obtain the following system of ordinary dierential equations in : ig () + G () = i[^f(;k)!+k( + K K )]G () +G ()( K) = "G () where F = U G and F = U G and ^F(; k) =8 : Perturbing G i and! about : h+ k (k y i )i jj < jj (3:5) (3:6) G i = G () i + G () i + G () i + (i=;;3;)! =! +! +! + (3:7) Far-Field Non-reecting Boundary Condition August, 995

14 The Incompressible Case 3 and substituting these perturbations into (??) we get the following zeroth and rst order problems in : The Zeroth Order Problem: ( ' () = g +^F(;k)g ( K)' = (3:8) where G () i = g i, g "' and i! = ". 8 : i = ^G The First Order Problem: g +^F(;k)G ( K) = ig h ^F(; k)! + K( + K K ) i (3:9) where ^G = "G () and = G(). Again, we need to match solutions at jj = (at the boundary of the wake region) when K =. Formally setting k = because out of the wake region the grid stretching should have no eect either on or on! we obtain the linear perturbation system in the free stream region: ( ig () + G () = i[!+]g ()="G () (3:) " G whose constant coecients imply a solution of the form: G = e r" ^G () ^G () (3:) Expanding as before, around ^G i =^g i +^G i + r = r +r +! =! +! + we obtain the following zeroth and rst order problems in the free stream region: th order ( ) : r ^g = (i! "r )^g = (3:) Far-Field Non-reecting Boundary Condition August, 995

15 Derivation of the Far-Field Non-Reflecting Boundary Condition from which immediately ^g =. Since i! = perturbations will not decay in. " then ^g = as well, otherwise the st order ( ) : i^g + r ^G + r ^g = ^G (i! "r )= (3:3) As in the zeroth order problem above, both ^G = and ^G =. This implies that the boundary conditions for the zeroth order problem are g ()= '()= and that the boundary conditions for the rst order problem are (3:) G ()= ()=: (3:5) Notice that the zeroth order problem reduces to: g +^F(;k)g = g ()= (3:6) Since < ^F(; k) < when k> then the Sturm Comparison and Oscillation theorems (see [?]) indicate that the minimal positive eigenvalue in (??) will be greater than the minimal positive eigenvalue in the non-stretched incompressible case. This fact is crucial and to a certain extent indicates why local time stepping is so eective. Since it is believed that indeed the long wave perturbations are the slowest to converge to steady state, then we have just shown that the decay of these disturbances is faster in local time stepping regimes than in global time stepping regimes. In (x5) we present numerical evidence indicating that the nonreecting boundary condition is quite eective in local time stepping regimes indicating that the absorbed long waves do in fact decay aswe predict. This is an interesting illustration of one of the reasons why local time stepping is eective. Derivation of the Far-Field Non-Reecting Boundary Condition Once the two eigenvalues, and! are found, an appropriate far-eld boundary condition can be developed to accommodate the physical situation at the far-eld. Far-Field Non-reecting Boundary Condition August, 995

16 Derivation of the Far-Field Non-Reflecting Boundary Condition 5. Derivation for Single-Grid Numerical Methods From the ansatz (??), we have: p = e i( t+bx) U G Substituting the dimensionless temporal and spatial variables: into the prole, we obtain (up to order ): p = e is l (! +! +) U G ()+p : l Taking a derivative with respect to and using the identity is! =, we arrive 9 at the far-eld = 9 (p : (:) In the incompressible case where =, the condition takes = 9 (p : (:). Derivation for Numerical Methods Utilizing Multigrid Acceleration The non-reecting boundary condition just developed was based upon an analysis around long waves. Should multigrid acceleration be used in conjunction with the numerical scheme then short waves must be taken into account aswell. Substituting P for p p, the boundary condition (??) may be = 9 : (:3) Taking the Fourier transform in x of the above equation we obtain which has the general solution: ^P t = 9 ^P + ik! ^P: (:) ^P = Ce ( 9 +ik! )t : For short waves, or high Fourier modes, k, the highly oscillatory behavour of the symbol ^P with t, will cause short wave disturbances in the numerical solution. This Far-Field Non-reecting Boundary Condition August, 995

17 Numerical Results 6 is especially critical when using multigrid techniques whose goal on each grid is to damp short waves. Our original nonreecting boundary condition, while eective in damping long wave disturbances caused by reections, may create new short wave disturbances and destroy the eects of the multigrid acceleration. By replacing ik in (??) with +ik we obtain a new Fourier symbol free of oscillatory behaviour even as k gets large: ik ^P = Ce ( The symbol (??) comes from the equation: 9 + ^P t = 9 ik ^P + +ik! ^P : ik +ik! )t : (:5) Transforming back to the original variables we obtain the boundary condition to be used with multigrid = 9 @t + 9 (p p ) (:6) = 9 (p p : (:7) 5 Numerical Results In this section we presentvarious numerical results obtained using the far-eld boundary condition just derived. We break this section into three subsections. The rst describes the numerical technique for calculating and! in compressible cases. The second section deals with the numerical solution of the incompressible boundary layer equations both with global and local time stepping. The third presents the results obtained in compressible viscous ows around the NACA airfoil using global time stepping regimes with and without multigrid acceleration. 5. The numerical calculation of and! in the global time stepping case The discretized form of the zeroth order problem (??,??) is a generalized eigenvalue problem: A~x = h B()~x (5:) Far-Field Non-reecting Boundary Condition August, 995

18 Numerical Results Omega K=.95 K=.8 K= Mach Number Figure 3: versus M for K =.7,.8,.95 where h is the cell size,, and ~x is (g ;g ;'; ). The problem is non-linear since B = B() is a function of, so the following iterative scheme was used: ( = A~x = h n+ B( n )~x (5:) where at each iteration, the smallest positive was determined. Generally convergence was obtained after three iterations and the main result is (Figure??): For all physically relevant values of ", M and K (see eq.??); =. The discrete rst order problem (??,??) can be written in the form: C() ~ X ( ~ A h ~ B()) ~ X = ~R +! ~R (5:3) where ~ X =(G ;G ;; ) and where A $ ~ A and B $ ~ B except that the boundary conditions on g are not parallel to those on G. The technique used to obtain! was to nd the left hand null vector of C(), ~v, and then! = (~v; ~ R ) (~v; ~ R ) : (5:) Unlike,! was not found to be constant for all physically relevant parameters. Figure?? graphically shows the values of! as a function of K for M = ; :; :; :3 (for neatness M =:;:5were not shown). The main result is: Far-Field Non-reecting Boundary Condition August, 995

19 Numerical Results omega for epsilon = M= M=.3 M=. M= K Figure :! versus K for " = : K!!! for <M:5and for all physical ". The limit M! is singular. In Table?? values of and! foravariety of parameters, ", M and K for time consistent numerical integration schemes are listed. These values were obtained through a program written in MATLAB. 5. Incompressible Viscous Flow 5.. Case - Global Time Stepping Regimes The viscous incompressible boundary layer equations (for a at plate) (??) have been solved in the geometry of Figure??. The Reynolds number (with respect to the plate length), R l, taken was, and a rectangular grid (76x6) with cell aspect ratio, x,of5was used. The plate had 76 nodes in the x direction, and y there were x nodes in the wake region. At inow, one quarter the way down the at plate, a Blasius prole was placed. The outow boundary was placed four platelengths from the trailing edge. The momentum equation was integrated in time using a Runga-Kutta scheme as suggested by [?], where the rst order spatial derivatives were discretized using a compact fourth-order scheme developed by Abarbanel et. al. [?]. The second derivatives were discretized using standard second-order central dierencing. Far-Field Non-reecting Boundary Condition August, 995

20 Numerical Results 9 K " M! Table : and! for various K, ", and M Far-Field Non-reecting Boundary Condition August, 995

21 Numerical Results Log(L norm of residual) extrap nonrefl Iterations Figure 5: Convergence behaviours for incompressible case. Two cases were run. The rst with extrapolation of u at outow, and the second using our far-eld boundary condition (incompressible, (??)) at outow with! updated every iteration as in (??). In each case 5 time steps were taken. Figure?? shows the convergence behaviour. It is evident that the non-reecting condition reduced markedly the residuals in the ow eld, implying that the the non-reeecting boundary condition produced a more stable numerical solution than did extrapolation. In order to test the eect of the nonreecting boundary condition as the outow boundary is moved closer to the trailing edge, additional runs were performed with the outow boundary at s = and at s = 3. Figure?? shows that as s decreases, the eects of the nonreecting boundary condition are more dramatic. In addition, Call S, S3, and S the solutions of the atplate problem on grids G G3 G respectively. Use N as an extention for a case with the nonreecting boundary conditions at outow, and use E as an extention for a case with extrapolation conditions at outow. We looked to see to what degree and SN SN3 SN; SE SE3 SE: It turns out that SN SN3 SN up to four signicant digits whereas SE SE3 SE only up to two signicant digits. Hence, we are certain that our boundary condition maintains accuracy better than the extrapolation condition does. All of this indicates that one could use a smaller computational domain while using the non- Far-Field Non-reecting Boundary Condition August, 995

22 Numerical Results x - 3 centerline outflow 6 8 x centerline outflow Figure 6: Residuals after 5 time steps - Extrapolation (top) Non-reecting (bottom) 6 8 reecting boundary condition and maintain \large computational domain" accuracy, thereby dramatically reducing the amount of computational work needed. 5.. Case - Local Time Stepping Regimes We shall now describe the results obtained using the nonreecting boundary condition in conjunction with a local time stepping regime. Let us state the conditions of the case studied. () The Grid: As in subsubsection 5.., the global time stepping case, the equations are discretized on a 76 6 rectangular grid. The inow boundary is located one quarter away down the at plate and the outow boundary is placed chord lengths down stream from the trailing edge. While x is constant, y grows geometrically so that y i;j = k j y i;, where y i;j y i;j y i;j. The expansion factor, k, was chosen so that in 6 intervals, y reaches one chord length. It should be noted that the aspect ratio of cells above the plate and the centerline are set identical to those in the global time step case so that in the local time stepping case t is determined by the parabolic part of the system in most of the wake region, recall (??). () Physical conditions: In this case, R l is again taken to be. Also, the boundary condition at inow is the Blasius prole one-quarter way down the plate. The outow boundary is located chord-lengths away from the trailing edge. Equations (??) and (??) with their boundary conditions, (??) and (??) are numerically Far-Field Non-reecting Boundary Condition August, 995

23 Numerical Results Ratio log(resid ext/resid nonreflect) after 5 time steps log(resid ext/resid nonref) after 5 time steps Distance of outflow boundary from trailing edge Figure 7: The ratio log nal residual with extrapolation bc nal residual with nrbc after 5 iterations as a function of the distance of the outow boundary from the trailing edge of a at plate. solved in this case for and! using a program written in Mathematica. The algorithm is similar to the the one presented in subsection 5.. In this case, is not a constant and depends upon the expansion factor, k. For these physical conditions we obtain =5:9597 which is signicantly larger than, obtained in the global time stepping case. This implies that the long wave disturbances decay faster when local time steps are used than with global time steps. Regarding!, recall that in the global time stepping scheme case the relationship between between! and K was linear. Therefore, the relationship between! and c d was linear as well. It has been numerically determined for the present local time stepping case, that the relationship between! and c d is linear as well, more specically: Far-Field Non-reecting Boundary Condition August, 995

24 Numerical Results 3 x - (a) x -6 (b) inflow centerline inflow centerline x -5 (c) x -6 (d) inflow centerline inflow centerline Figure 8: Residual plots after 5 iterations (a) extrapolation, gts (b) non-reecting condition, gts (c) extrapolation, lts (d) non-reecting, lts! =56:3c d :3 : (5:5) The non-reecting boundary condition with these coecients was used as an out- ow boundary condition, as was extrapolation. The results were compared and we state them now. In gure (??) the pointwise residuals after 5 iterations for the four test cases (extrapolation outow conditions{global time stepping, non-reecting outow conditions{global time stepping, extrapolation outow conditions{local time stepping, and non-reecting outow conditions{local time stepping). are plotted. In gure?? the L norm of the residuals (for each case) is plotted as a function of time steps. Figure?? (a) graphs the logarithm of the L norm of the residual as a function of time steps (iterations) for global time stepping runs. Figure?? (b) graphs the logarithm of the L norm of the residuals as a function of time steps for local time steping runs. The bottom dotted line is a special case that will be discussed later on in this section. It is clear from the results that Local time stepping regimes converge faster than their global time stepping counterparts. The case with non-reecting outow conditions with global time stepping reaches steady state faster than the case with extrapolation outow conditions and local Far-Field Non-reecting Boundary Condition August, 995

25 Numerical Results - (a) - (b) - -3 extrap... nonref - -3 extrap... nonref log average residual log average residual iterations iterations Figure 9: Residuals (a) global time stepping (b) local time stepping time stepping. Note that adding the non-reecting boundary condition to the local time stepping scheme speeds up the convergence. This implies that the non-reecting boundary condition accelerates convergence to steady state above and beyond local time stepping. For experimental purposes only we ran the at plate code with local time stepping and the non-reecting boundary condition using the appropriate and! for global time stepping schemes. The result is that the convergence behaviour in this case is unexpected. In fact, the convergence is better than with local time stepping with the non-reecting condition along with it's appropriate and!. The bottom dotted line in (??) shows the convergence behaviour in this case, and another order of magnitude is obtained. Figure (??) shows the pointwise residuals after 5 iterations in this additional case. Clearly the best steady state is achieved in this way. This in no way disproves our claim. We obtain better convergence using our non-reecting boundary condition than with extrapolation conditions both in global and local time stepping regimes. However, it appears that our method of nding the coecients is not optimal, so that perhaps factors other than long wave disturbances need to be taken into consideration. In any case, the important result that we have demonstrated here is as follows: Local time stepping schemes converge to steady state faster than global time stepping schemes do because they cause longwave disturbances from Far-Field Non-reecting Boundary Condition August, 995

26 Numerical Results 5.5 x inflow Figure : Local time stepping run with and! values. Residuals 5 3 centerline taking on global time stepping steady state to decay faster than schemes utilizing global time stepping. 5.3 Compressible Viscous Flow around airfoils 5.3. Case - Global time stepping without multigrid acceleration The far-eld non-reecting boundary condition was implemented in the program [?], as a means to test its eectiveness in the compressible viscous subsonic case with R l = 5 and M =:5. An appropriate C-mesh was constructed using ahyperbolic grid generator. Often, hyperbolic generators create grids that tend to magnify the intrinsic singularity at the trailing edge. By adding additional control points we haveovercome this phenomenon and generated a smooth grid reminiscent of an ellipticly generated one. The outow boundary was placed 5 chord lengths away from the trailing edge, and was made as perpendicular as possible to the airfoil chord. Two runs were made, one with extrapolation boundary conditions and the other with the nonreecting boundary condition. In each run, 3 time steps were taken. Neither multigrid nor other accelerators were used. Minimal articial viscosity was used to assure convergence. The results that have been obtained are delineated below. The L and L norms of the residual are shown in Figures?? and??. Clearly the nonreecting condition accelerates convergence to steady state more than the Far-Field Non-reecting Boundary Condition August, 995

27 Numerical Results 6 Residuals -- Maximum Norm - log(max density residual) extrap nonrefl iterations x Figure : The L norm of the residual extrapolation condition does. The peaks in the residuals while using extrapolation have not yet been explained. Due to stability considerations (long time numerical integration), in both cases the residuals increase and level o towards the end of the runs, but in the nonreecting case, the smallest residuals are smaller than in the extrapolation case, and the residuals grow sooner and faster with extrapolation than with the nonreecting condition. Hence, the nonreecting condition produces a more robust computational environment, and achieves a better steady state. In Figures?? and??, the pointwise residuals are shown for extrapolation and nonreecting boundary conditions for 3 time steps. While 3 time steps is clearly beyond the optimal computation time, it is still evident that the nonreecting boundary condition is more eective at reducing residuals than the extrapolation condition. In addition, in Figure?? the values of and u along the grid line: j =5, i= 58 are shown. This grid line starts in the wake region (far eld), then is in the boundary layer (around the airfoil), and returns to the far eld. Both conditions give identical results around the airfoil { in the boundary layer. However, in the far eld, the nonreecting condition produces a smoother solution than extrapolating does. While we have shown this result for j = 5 only, it is true in the entire wake region. Far-Field Non-reecting Boundary Condition August, 995

28 Numerical Results 7 - Average Residuals - log(average density residual) extrap nonrefl iterations x Figure : The L norm of the residual 5.3. Case - Global time stepping withmultigrid acceleration In order to decrease the residual further and to stabilize the runs, wehaveinvestigated the nonreecting boundary condition in regimes with global time steps, accelerated to steady state with multigrid techniques. As we previously mentioned, the nonreecting boundary condition needs to be modied when using multigrid acceleration in the numerical scheme and it takes the form (??). In this case we compared two runs { one run using multigrid acceleration and extrapolation outow conditions, and the other run using multigrid acceleration and the above nonreecting boundary condition. In order to help eliminate short wave disturbances in the y direction, we further introduced the Fourier smoothing as suggested by Ryaben'kii [?] p n i;j pn i;j+ + pn i;j + pn i;j (5:6) locally near the lower and upper corners along the outow boundary This smoothing transformation was applied each time our nonreecting boundary condition was called and can be shown to identically kill o the shortest wave in the Fourier expansion. In the gures which follow (gures??,?? and??), we show the convergence behavior in both the L and L norms in using at outow extrapolation and the nonre- ecting condition and show residual plots for for both the extrapolation boundary condition and the nonreecting boundary condition. Far-Field Non-reecting Boundary Condition August, 995

29 Numerical Results 8 x -5 x (a) (b) x - x (c) (d) Figure 3: Residuals using Extrapolation conditions for (a), (b) u, (c) v and (d) e The same C-grid and other ow parameters were used in these calculations as in the previous subsection. Multigrid acceleration was applied in the following way: W cycles (3 grids) were run rst starting on the rst coarser grid in order to set up a reasonable initial condition. The results of these sweeps were used as the initial condition for a set of W cycles ( grids) starting on the nest grid. Both second order and fourth order articial viscosity (see [?] and [?]) were applied as well as implicit residual smoothing. Residual smoothing insures that the scheme will converge with a much larger CFL number hence accelerating convergence. The results are as follows: The L residual norms obtained using both boundary conditions are nearly identical. However the residual plots indicate that the largest residuals at steady state with the both outow conditions are at the corners of the outow boundary (the corner problem) and at inow, presumably propagated backward from the corners. The maximum norm, L of the steady state residual while using the nonreecting boundary condition at outow, is slightly greater then when using the extrapolation conditions at outow as is conrmed in gure??. However, in the wake region, the residuals are smaller when using the nonreecting boundary condition then when using extrapolation conditions. This is clearly seen in gures?? and??. Far-Field Non-reecting Boundary Condition August, 995

30 Conclusion 9 x -6 x (a) (b) x -6 x (c) (d) Figure : Residuals using Nonreecting conditions for (a), (b) u, (c) v and (d) e 6 Conclusion We have described the derivation of a nonreecting far eld boundary condition for steady viscous, compressible, two dimensional external ows. This condition produces a smoother far eld solution than the presently used extrapolation boundary conditions, while retaining the solution in the boundary layer. It also smooths out the trailing edge singularity by one to two orders of magnitude. The method can be applied to incompressible ows, where it is just as eective in driving residuals down to steady state faster than extrapolation conditions. In addition, nonreecting boundary conditions of the type described in this work can be developed for schemes whose time stepping is local, as well as for schemes utilizing multigrid acceleration. In addition to the benets accrued from using out boundary condition, it is easy to program, and does not require the use of much additional memory in an existing program. Also, its simplicity and local form require little cpu time. Perhaps the most important feature of our approach is that while in this paper we have chosen to deal with steady state problems, our methodology could be used to develop a condition for evolution equations that do not reach a steady state. In this case, we would perturb the linerized equations about the appropriate wave group that we wish to absorb (not necessarily about long waves) and proceed exactly as we did for steady state ows. Of course an appropriate far eld prole needs to be formulated as well for each ow, which now may be time dependent. Hence our methodolgy is Far-Field Non-reecting Boundary Condition August, 995

31 Conclusion 3.5. nonrefl... extrap.5. nonrefl... extrap.5.3 density u velocity the section j=5, i= to the section j=5, i= to 58 Figure 5: The values of and u along the grid line: j =5,i= 58 versatile and is shown to be eective in reducing residuals in external viscous ows. References [] S. Abarbanel, A. Bayliss and L. Lustman, Non-reecting Boundary Conditions for the Compressible Navier-Stokes Equations, Proc. of the 7th International Congress in Computing Methods in Applied Science and Engineering, North Holland, (986). [] S. Abarbanel, A. J. Kumar,Compact High Order Schemes for the Euler Equations, J. Scientic Computing, Vol. 3, (988), [3] G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, (967). [] G. Birkho, G.-Carlo Rota, Ordinary Dierential Equations th ed., Wiley, (989). [5] B. Gustason, H.-O. Kreiss, Boundary Conditions for Time Dependent Problems with an Articial Boundary, J. Comp. Phys., 3, (979), [6] A. Jameson, H. Schmidt, and E. Turkel, Numerical Solutions of the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes, AIAA Paper No.8-58, (98) Far-Field Non-reecting Boundary Condition August, 995

32 Conclusion 3-3 log(average residual) - log(max residual) (a) (b) Figure 6: The L, (a), and L,(b) norms of the residual after multigrid W cycles. Legend: - nonreecting conditions, - extrapolation conditions [7] R. D. Richtmeyer, K. W. Morton, Dierence Methods for Initial Value Problems, nd ed., Interscience, (967). [8] Private Communication. [9] R. C. Swanson, Eli Turkel, Articial Dissipation and Central Dierence Schemes for the Euler and Navier-Stokes Equations, AIAA CFD Conference, AIAA paper No cp, (987). [] Eli Turkel, Numerical Methods for Large-Scale, Time-Dependent Paritial Differential Equations, Computational FLuid Dynamics Vol, W. Kollmann ed., Hemisphere Publication Co., (98), 7-6. [] Eli Turkel, Improving the Accuracy of Central Dierence Schemes, th International Conference on Numerical Methods in Fluid Dynamics, Springer-Verlag Lecture Notes in Physics, Vol 33, (988), [] Private Communication. Far-Field Non-reecting Boundary Condition August, 995

33 Conclusion 3 Density Residuals- Multigrid.5 x Figure 7: The residuals after multigrid W cycles. Extrapolation outow conditions Density Residuals- Multigrid.5 x Figure 8: The residuals after multigrid W cycles. conditions. 6 Nonreecting outow Far-Field Non-reecting Boundary Condition August, 995