A Finite volume simulation of a supersonic laminar Flow: Application to a compression corner

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1 Journal of Materials an Environmental Sciences ISSN : opright 07, Universit of Mohamme Premier Ouja Morocco JMES, 07 Volume 8, Issue, Page Introuction A Finite volume simulation of a supersonic laminar Flow: Application to a compression corner D. Bahia,, N. Salhi 3. Department of Phsics, Multiisciplinar Facult of Naor, Universit of Mohamme Premier Ouja, Morocco. Laborator: Observatoire e la Lagune Marchica e Naor et Régions Limitrophes (OLMAN-RL) 3. Department of Phsics, Facult of Sciences of Ouja, Universit of Mohamme Premier Ouja, Morocco 4. Receive 6 Jan 07, Revise 3 Ma 07, Accepte 4 Ma 07 Kewors Navier-Stokes; Numerical analsis; Supersonic flow; Finite volume; ompression corner D BAHIA risbahia97@ahoo.fr Abstract In this paper, the finite volume metho is use for the computation of a supersonic laminar flow. As an application, we consier the flow over a compression corner with ifferent angles. The invisci flues are approimate using Steger an Warming Flu Vector Splitting metho with the MUSL approach to increase the spatial accurac. The transport properties (viscosit an conuctivit) which are functions of two inepenent thermonamic parameters (T, P), are calculate using a centere scheme. The performance of the metho that is use here is shown b some comparisons with other results. Viscous interaction is one of the major aspects of supersonic flows.the interaction between a shock wave an a bounar laer can prouce a region of separate flow []. Mac ormack metho was one of the most popular of the central schemes []. Space-centere schemes of secon orer accurac in space where initiall introuce with implicit linear multi-step time integration methos b Bril an McDonal [3], an Beam an Warming [4]. Due to the oscillations that are create in central algorithm, Up win schemes were introuce to prevent these oscillations. The first eplicit Up win scheme was introuce b ourant, Isaacson an Reeves (95), since this ate there has been consierable effort aime at constructing an analzing high orer accurate, non-oscillator approimations to hperbolic conservation laws [5-6]. First Van Leer publishe his work on the "Ultimate ifference schemes" in a series of traitional papers [7-]. This work represente a complete etension of the iea of Gounov to high orer schemes. urrentl, one of the most popular solvers is ue to Roe [], who propose a linearize Riemann solver for the Euler equations, where the thermonamic properties are represente b the perfect gas law. Since 98, several generalizations of the Roe solver were propose in orer to inclue the effects real gases [3-5]. More recentl, Gallout an Masella [6], presente a numerical scheme, VFRoe, of finite volume tpe for the sstem of nonlinear one-imensional hperbolic equations. Flat plate an compression corner flows appear in man supersonic aircraft configurations an particularl interesting feature is the interaction of a bounar laer with shock waves. Within this framework several works was complete knowing that the work of arter J. [7] is the reference of all work since 97. Rizzetta D. [8]; Tenau. [9]; Mallinson S. G. [0]; Fang J. []; Jammalamaaka et al. []; Yang G., Yao Y., Fang J., Gan T. [3] are consiere as members of the same subgroup who treat the compressible supersonic flows on a compression corner with ifferent methos. In this work, a secon orer finite volume metho is use for the resolution of a supersonic laminar flow over compression corner, on a general quarilateral meshes. The paper is organize as follows: the governing equations are given in section ; section 3 escribes the finite volume approach an the numerical flu approimation. The numerical tests an comparisons with others results are showe in section 4. Bahia an Salhi, JMES, 07, 8 (), pp

2 . Presentation The compressible viscous flow is escribe b the conservation laws of total mass, momentum an energ. This sstem of conservation equations can be written in a general integral form, which is the basis of the finite volume metho, escribe later []. For an arbitrar control volume Ω, surroune b the surface S, the equations write U + F.S + F.S = 0 S S t () Where U is the vector of conservative variables, F an F are the flu vectors of the invisci an viscous contributions, respectivel. ρ ρv 0 V U = ρv F = ρv V + P I F = ρe ρhv.v - k T is the velocit vector, ρ is the mass ensit, P the pressure, T the temperature, E the total energ per mass an k is the thermal conuctivit. The components of the shear stress are given b v v i j ij = ( + ) - (.V)δ ij 3 δ ij is the namic viscosit of the flow, is the smbol of Kronecker. To close this sstem, the equation of state for the pressure has to be ae: γ P j T is the ratio of specific heats. The variables in equations () an () ma be nonimensionalize as follows: = = L L u v t.l u = v = t U U U P = P = T = p U U E E = μ = μ U μ i () ct Where: is the specific heat at constant pressure, L is the length of the plate. U is the reference velocit, ρ is the c p reference ensit, μ is the reference viscosit an is the free stream smbol. The resulting sstem of nonimensionalize ifferential equations ma be convenientl as: U + F S.S + F S.S = 0 t (3) ρ ρ V 0 U = ρ V F = ρ V V + P I F = Re,L ρe ρ H V.V - μ Pr T Bahia an Salhi, JMES, 07, 8 (), pp

3 With: ρul is the Renols number, Re,L =, Pr is the Prantl number. μ An: v v i j γ - ij = ( + ) - (.V )δ ij P = T 3 c γ Re,L j i p 3. Finite volume iscretization The basic iea of a finite volume metho is to use a finite set of control volumes to escribe the equation an restrict the unknown to be in a finite imensional space [4]. Within this approach the integral conservation laws are written for a iscrete D control volume (ABD), calle cell (i, j) (figure ). Equation (3) is replace b the iscretize form: (U ABD. ABD ) + (F c + F ).S = 0 (4) t sies Where the sum of the flu terms refers to all the eternal sies of the control cell (ABD). We woul ientif U with with the area of (ABD). ABD U, ABD U = - (F c + F ).S = 0 (5) t S j ABD 4 faces j D ij B A S i The evaluation of the term the time accurac. U t i Figure : Two imensional control volumes in equation (5) etermines whether the scheme is eplicit or implicit an fies 3. onvective flu There are several was to carr out the convection flu [5-6]. Here we present an upwin scheme base on the "Flu Vector Splitting" metho of Steger an Warming [7]. The iea is to seek the characteristic lines of the flow on which the phsical information is propagate an to iscretize the flu accoring to this irection. The flu is ecompose into a positive part an a negative part accoring to the irection of the transport on these characteristic lines. The positive flues are evaluate using available information in the cells upstream of the normals = S.n, an vice versa for the negative flues. This splitting of the convective flu is base on the sign of the eigenvalues of the jacobian A.n we write: Bahia an Salhi, JMES, 07, 8 (), pp

4 - F c.n = A.n U = P(n).D(n).P (n) U (6) sie sie sie Where P(n) an D(n) are respectivel, the transformation matri an the iagonal matri. The eigenvalues of the Jacobian matri of the sstem are: = = V.n; = V.n + c; = V.n - c Where c is the spee of the soun: 3 4 γp c = We ecompose the iagonal matri D as: D = D + + D - The matri D + contains onl the positive terms What gives: j an vice versa for the matri D - : +,- j = ( j j ) = (γ - ),,, F c.n = P(n).D(n).P (n) U = P(n).D (n).p (n) U sie sie sie - - P(n).D (n).p (n) U = F c.n F c.n sie sie sie In the D the following epressions are obtaine from the split convective flues: where: = (γ - ) + +,, u cn, ( 3 4 ),, F c.n = v cn ( 3 4 ) sie γ u v c (,,,, 3 4 ) c(v.n)( 3 4 ) γ- +,- +,- +,- 3 4 n = (n, n ) is the outwar unit normal vector of the calculation face. The spatial accurac The simplest upwin scheme results if a zero-orer etrapolation of the conservative variables is use: F c.n = F c.n i +, j i, j F c.n = F c.n i +, j i +, j The resulting scheme is onl first-orer accurate in space. Higher orer schemes can be obtaine using a linear etrapolation of the conservative variables. This linear etrapolation is performe b using a quasi-d approach. For eample, the conservative variables on the bounar surface (i+/, j) will be obtaine b etrapolating the conservative variables in the noes (i-,j), (i, j), (i+, j) an (i+, j). The conservative variables use in the positive split flues are enote U whereas U values use in the negative split flues. The following etrapolation formulas are use: U AB = U + (R ) U - U i-,j i-,j U AB = U i+,j + ( ) U i+,j - U i+,j R Bahia an Salhi, JMES, 07, 8 (), pp i+,j i+,j i+,j i+,j i+,j (7) (8) stans for the (9)

5 Where the function is efine as: R with: - k + k (R ) = + R is the ratio of the graients ownstream an upstream of the cell (i, j) : U - U - R = U - U - i+,j i-,j i-,j i+,j = - i, j + - i, j i+, j i+, j The etrapolation ensures that the scheme is of secon-orer accurac in space. The parameter efines ifferent schemes. The most values currentl use are k=- (full upwin scheme), k=0 (partiall upwin scheme). It is clear that the higher orer schemes are not monotonous. This ma lea to oscillations into the solution especiall in the vicinit of large graients such as shocks. In orer to restore monotonicit limiters which are introuce in the etrapolation formulas. This leas to a reefinition of the function : The following limiter of Minmo is use: - k + k (R ) = + R Lim 0 if R 0 Lim = R if 0 R if R (0) 3. Diffusive flu Because of the elliptic character of the iffusive flu [] (the phsical information is propagate everwhere), the viscous an thermal iffusion terms are compute here b a secon-orer centere scheme. First, the flu is ecompose in the two irections as follow with : (F ) = (F.n ) + (F.n ) () 4 faces 4 faces 4 faces 0 u v ( - ) 3 F = - u v ( - ) 4 u v u v T u - u + v + v + k u v ( + ) F = - v u ( - ) 3 4 v u u v T u - v + u + u + k 3 3 The erivatives acquiring in the epressions for corner noes of the cells. F an F are foun b averaging the mean erivatives in the Bahia an Salhi, JMES, 07, 8 (), pp

6 B 4 3 D A Figure : The mesh for the iscretization of the iffusion flu We calculate the graient on the face AB of Figure with: u u u + AB A B Then we compute the graient at point A as the mean value of the one on the surroune cell (34): u u = u ABD S 34 AB A similar relation is foun for the erivative with respect to : ( u u )( ) ( u u )( ) ( )( ) ( )( ) u u = u ABD S34 AB ( )( u u ) ( )( u u ) ( )( ) ( )( ) (3) () (4) 3.3 Time integration We use a first-orer eplicit scheme for the iscretization in time; therefore the Eq. (3) becomes: n+ n t U = U - (F c.s) + (F.S) ABD 4sies 4sies Stabilit analsis (5) The scheme (5) escribe above is applie to the D linear convection-iffusion equation: u u u u u + a + = + t is the kinematic viscosit of the flow. Neumann analsis on the stabilit of the scheme escribe before give the following stabilit conition: where t is time step, an an t 0.5 Pr Re,L γ + are the spatial steps. 4. Results an iscussion: Supersonic flow over a compression corner with angles of 5, 7.5 an 0 In this Navier-Stokes problem, a Mach 3 flow passes over a compression corner at an angle of 5, 7.5 an 0. The Renols number, base on the free-stream values an the istance from the leaing ege of the flat to the (6) (7) Bahia an Salhi, JMES, 07, 8 (), pp

7 corner, is The geometr of the phsical omain an the wa of gri use are shown in the figure 3 below. M u.0 v 0.0 T /(( ) M) the same conitions to the Upstream uv0.0 free surface 5, 7.5, 0 Upstream conitions Wall conitions: ; Twall T0, ( M ) ( ) M Figure 3: The Mach 3 flow over compression corner with angles of 5, 7.5 an 0 The computational omain covers the area 0.0, on the plate, an a height of 0.575above the wall past the corner. The leaing-ege of the plate is place at = 0 an the corner at =. The total number of gri points use was 56, thereb a constant gri spacing in the an irections. Figure 4 shows the wall pressure istribution for ramp angles of 5, 7.5 an 0. It is seen that the pressure is a ecreasing function on an it is an increasing function on 0.7. The profiles possess c c iscontinuities at the corner =. c Figure 4: omparison of wall pressure istribution for Ramp angles of 5, 7.5 an 0 Figure 5: omparison of the isplacement thickness measure normal to the surface istribution for ifferent ramp angles Bahia an Salhi, JMES, 07, 8 (), pp

8 Figure 5 presents the isplacement thickness istribution. The profiles are compose of two parts, the first part is an increasing function an the secon one is a ecreasing function. The profiles possess a maimum at the corner. This maimum value is more important for ramp angle 0. Figure 6: omparison of wall skin-friction istribution for ifferent ramp angles Figure 6 presents the skin-friction istribution. Here also, the profiles are compose of two parts, the first part is a ecreasing function an the secon is an increasing function. The profiles possess a minimum at the corner. This minimum value is more important for the ramp angle 0. These results show goo agreement with the results of arter J. [7], Fang J. [] an Babinsk H. [3]. Figure 7: omparison of the results of present calculation an the results of arter J. an Hung & Mac ormack Figure 7 presents the profiles of the friction coefficient, heat-flu coefficient an pressure coefficient along the wall, the results of present calculation show goo agreement with the results of arter J. [7], Hung. M., Mc ormack R. W. [6] an Jammalamaaka A. []. Graph of Densit Graph of Pressure Graph of Temperature Figure 8: Solution of the Mach 3 compression corner problem Figure 8 shows the profiles of aiabatic an isotherm contours, this figure emonstrates the abilit of the metho to capture the leaing-ege shock, the results of present calculation show goo agreement with the results of arter J. [7], Yang G. [3], WU M. & Martin M. P. [8], Babinsk H. & Harve J. K. [9] an Zapragaev V. I., Kavun I. N. & Lipatov I. I. [30]. The number of ccle require for convergence of the calculations iscusse was 500. However for the calculation carrie out b J. arter there is 500 to Bahia an Salhi, JMES, 07, 8 (), pp

9 onclusions This paper presents a numerical simulation of the Navier -Stokes equations that govern a supersonic laminar flow over a compression corner. An upwin secon orer finite volume scheme is use for the iscretization of the convective flues, on general quarilateral meshes, whereas the iffusive contributions are compute with a centere scheme. The results of the test case consiere here an the comparisons mae with the results of Fang J. [], Jammalamaaka A. [], Yang G. [3], WU M. [8], Babinsk H. [9] an Zapragaev V. I. [30] show the goo performance of the algorithms emploe. References. Hirsch., Vol.. (990).. Mc ormack R. W., J. AIAA. 4th Aeronamic Testing onference. 4 (969) Brile W. R., Mc Donal H., Proc. of the 4 th Int. onf. on Numerical Methos in Flui Dnamics. 35 (975) Beam R. M., Warming R. F., J. omp. Phsics. (976) Harten A., J. Publ Res I Math Sci. 7 (987) Boris J. P., J. omp. Phsics. (973) Van Leer B., Springer Lecture Notes in Phsics. 8 (973) Van Leer B., J. omp. Phsics. 8 (974) Van Leer B., J. omp. Phsics. 3 (977) Van Leer B., J. omp. Phsics. 3 (977) 76.. Van Leer B., J. omp. Phsics. 3 (979) 0.. Roe P. L., J. omp. Phsics. 43 (98) ollela P., Glaz H., J. omp. Phsics. 59 (985) Glaister P., J. omp. Phsics. 74 (988) Liou M., Leer B. V., Shuen J., J. omp. Phsics. 87 (990) Gallouet T., Masella J. M.,. R. Aca. Sci. Paris. 0 (996) arter J. E., NASA TR R-385 (97). 8. Rizzetta D. P., Visbal M. R., Gaitone D. V., J. AIAA. 39 (00) Tenau., Garnier E., Sagaut P., Int. J. Numer. Meth. In Fluis. 33 (000) Mallinson S. G., Gai S. L., Mufor N. R., J. Flui. Mech. 34 (997).. Fang J., Yao Y., Zheltovoov A., Li Z., Lu L., J. Phs. Flui. 7 (05) Jammalamaaka A., Li Z., Jaberi F., J. Phs. Flui. 6 (04) Yang G., Yao Y., Fang J., Gan T., Li Q., Lu, L., hinese Journal of Aeronautics (JA). 9 (06) Hirsch., Vol.. (988). 5. Pironneau O., J. Numer. Math. 38 (98) Hung. M., Mc ormack R. W., J. AIAA. 4 (976) Steger J. L., Warning R. F., J. omp. Phsics, 40 (98) WU M., Martin M. P., J. AIAA. 45 (007) Babinsk H., Harve J. K., ambrige Universit Press (0). 30. Zapragaev V. I., Kavun I. N., Lipatov I. I., Progress In Flight Phsics. 5 (03) 349. (07) ; Bahia an Salhi, JMES, 07, 8 (), pp