Flow problem in three-dimensional geometry

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1 UDC Flow problem in three-dimensional geometry Maxim Zaytse 1), Vyachesla Akkerman ) 1) Nuclear Safety Institute of Russian Academy of Sciences Moscow, ul. Bolshaya Tulskaya, 5, Russia ) West Virginia Uniersity Morgantown, WV , USA Abstract It is shown how a complete set of hydrodynamic equations describing an unsteady threedimensional iscous flow nearby a solid bod can be reduced to a closed system of surface equations using the method of dimension reduction of oer-determined systems of differential equations. These systems of equations allow determining the surface distribution of the resulting stresses on the surface of this body as well as all other quantities characteriing the hydrodynamic flow around it. Keywords: flow problem, hydrodynamic flow, iscosit solids, stress on the body surface, differential equation on the surface 1

2 Introduction In application, nonlinear equations of hydrodynamics are often one of the obstacles inoled in the study of many phenomenas, such as turbulence, hydrodynamic discontinuities motion, moement of the fronts of chemical reactions, etc. [1-4]. (During their modeling in practical problems often one has to produce a ery fine numerical grid in space and time, which requires large computing power and time-consuming) When modeling such equations in practical problem, one has to often produce a ery fine numerical grid in space and time which is time consumming and requires large computing power hence rendering the methode as not practical [5]. In this regard, arious ways of reducing the complete system of hydrodynamic equations in terms of a system of surface equations are of a large scientific interest. [6-11]. Such a procedure can reduce the dimension of the problem by one (3D D, D 1D ), which significantly reduces the required computational power. In particular, the corresponding computer program could directly calculate the hydrodynamic discontinuities with iscosit sound formation and other changes in the density of gases and liquids (using the information only on the surface). For example, the problem of describing the potential flow in a plane can be reduced to an integral equation on the boundary for a anishingly small iscosity (Dirichlet, Neumann) [1, 4]. This equation relates the tangential and normal components of the elocity. If one of the components is known on the boundary of the bod one can define all the external flow. In this paper we reduce the equations of a iscous medium "in the olume" to surface equations and demonstrate how the proposed method can reduce the dimension of the problems in iscous flow, which makes the equations easier to simplify the calculations in a ariety of applications. General methods proposed for oerdetemination of the Naier-Stokes equations in three-dimensional and two-dimensional case, which theoretically allows reducing them to a closed system of equations on any surface [1, 13]. No significant simplifying assumptions, which would limit the generality of the consideration of these phenomena, are considered. 1. Formulation of problem Moement of solids in iscous media is characteried by the conditions on the boundary of a solid, where the particles stick to it. In other words, the rate of iscous medium on the surface of the body u is determined by the kinematic characteristics of the surface, for example,

3 it can be expressed in terms of the speed of the body surface V. Force F acting on a unit surface area is [1] F Pn n (1) i i / ik k where n is normal to the surface, and u i uk / ik () xk xi the iscous stress tensor and - coefficient of dynamic iscosity. Therefore, according to equation (1), in this case, for any point on the body surface in a Cartesian coordinate system 1, n, τ τ one has u1 u F n P, (3) 1 u1 u 1 n un F, (4) n 1 1 u u n u n F 1 (5) n (see Fig. 1). In addition, the force F relates to the elocity of the body V ia the laws of rigid body motion dynamics. Unsteady flow around fixed solids in iscous medium is different from the aboe by cancelling elocity conditions at the boundary [1]: V и u. (6). Oerdetermination of Naier-Stokes equations We confine ourseles here to an incompressible iscous flow in three-dimensional space in the graity field g. To illustrate this, the Naier-Stokes equations take the form [1] u 1 u P u g, (7) t u, (8) diu, (9) where - is the kinematic iscosity coefficient. These equations allow us to find the unknown elocity ector u r,t and the pressure profile P r,t at any gien point and time in space, i.e. for any r,t. We introduce a "correction" α to the ector u and pseudo-density r,t in order to generalie the system of equations (7) - (9) as follows 3

4 u α 1 P t u α α u g, (1) t α t u α diu α α u α α α, (11), (1) u, (13) diu. (14) Actuall the equations (11) and (1) respectiely determine r,t and α but the equation (1) is resulting from Equations (7) and (1). Vector field u α can be formally considered as 1 1 u u α. We go one to one to the pseudo Lagrangian ariables of initial position of the gas particles iniscid hydrodynamic flow with density r,t and pressure P and time ( r (i.e. marking) dr u(,) r t α (,) r t (,) r t α (,) r t, dt r r ( r и r r ( r, t), r t r. (15) We chose (,) r t so that α α. (16) Hence, it can be shown that r u α α r. (17) t Then, according to appendix A, from the equations (1) and (11) one yields the following expressions: αr r α r r 1 ( x, y, ) ( x, ) 1 r, (18), (19) where α,, u - initial distribution of the ariables α,,. Let α and 1. Then (18) and (19) can be transformed to the form [14]: r r x x r, () y 4

5 r r r y y, (1) x r r r. () x y Actuall from three equations () - () only two are independent [14]. Formula (19) takes the form 1 ( x, ) 1. (3) ( xy,, ) Thus, we hae obtained an oerdetermined system of 16 differential equations (7) - (9) (11) (1) (16) () (1) (3) plus any of the equations from (17) and of 15 unknowns α, u,, P,, и r. Causation analysis shows that the system is consistent and alid [1-14]. It is possible to add to the equation (1) a nontriial external force. 3. Viscous flow in a three-dimensional case We derie a system of equations describing the unsteady flow around the fixed rigid body in three-dimensional incompressible iscous stream. Let us denote S S1, S y, S3, S4 ux, S5 uy, S6 u, S7 x, S8 y, S9, 1 x P, S11, S1, S13 x, S14 y, S15. (4) Then, a system of 16 equations (7) - (9), (11), (1), (16), (), (1), (3) plus any of the equations (17) can be written in form H k S S S,,..., , k (5) r t We now turn to the coordinate system τ1, τ, n at point M (see Fig. 1). Then equations (5) can be written as S S S S Hk S,,,,..., , k (6) 1 n t We differentiate equations (6) in the direction n 14 times. Then we obtain the 4 equations of the form Hk S We denote i S S S S,,,... 1 n t n, , k , i (7) 5

6 S S n, (8) j j j where j...15, O h S S. Then 4 equations (7) can be written as: j j j j S S S S,,,..., , h 1...4, j (9) 1 t We extend the ariable S S along the boundary of the body as follow: 1 1 α n α n. (3) Let g ( r be the equation of the solid surface in the coordinate system ( r (15). Then from (6) and (3) its speed is equal to V g g t t g g u α α g g V un α n α n. (31) g In the coordinate system ( r the solid body is not moing. Thus, the dependence of the deriatie to n in the transformed system of differential equations (9) is missing, and we obtain a closed system of 4 surface differential equations strictly along the boundary of a solid and a corresponding number of ariables. In fact, some of the ariables are determined from (6) and (3). At the initial time, it is required to know nontriial orticity r and some of its spatial deriaties on the surface (formula (), (1)). It is always possible to find where the orticity is non-triial, because there is a iscous boundary layer on the surface. The adantage is that the system actually describes the process of flow past the body surface in terms of the surface itself. Calculations show [15, 16] that the expressions (9) obtained in an analytical form are extremely cumbersome. Their writing out is beyond the scope of this paper. Howeer, one proides a detailed algorithm of their obtaining. 4. Oerdetermination of the complete system of hydrodynamic equations Complete system of equations describing the hydrodynamic enironment in a nonuniform unsteady 3D case is [1] di t u, (3) 6

7 u u P F u, (33) t u, (34) =diu, (35) s T us Q diq, (36) t T T, s P P, s, (37), P w T s, (38) where, u, P, T, w и s - are densit elocit pressure, temperature, specific heat function, and the entropy (per unit mass) of the medium, respectiely. Here F и Q - are olumetric force and heat generation respectiel q T - is the heat flow equation ( - thermal conductiity), - the iscous stress tensor, for which we hae 1 4 u 3 u 3. (39) Here is - shear dynamic iscosity. The dissipation function in (36) is equal to ui uk ul 3 xk xi xl. (4) Equations (37) and (38) are thermodynamic relations that depend on the properties of the medium. We introduce a "correction" α r,t which we determine from equations: t α t u α diu α α u α α α to the ector ( r, t ) u and pseudo-density r,t,, (41) F Ts. (4) Termwise folded formula (4) and (33) and taking the operation on both sides, we find, using (38), that equation (33) is equialent to the following expressions u α t x u x x u α α w x, (43) 7

8 α u α α. (44) t We go one to one to the pseudo Lagrangian ariables of initial position of the gas particles and time ( r (i.e. marking) dr u(,) r t α (,) r t (,) r t α (,) r t, dt r r ( r и r r ( r, t), r t r. (45) We chose (,) r t so that α α. (46) Hence, it can be shown that r u α α r. (47) t Equations (41) and (44) can be transformed to the form (see Appendix A) αr r α r r, (48) 1 ( x, y, 1 ( x, ) ) r, (49) where α,, u - initial distribution of the ariables α,,. Let α and. Then it can be shown [14], equations (48) - (49) can be transformed to r r x x r, (5) y r r r y y, (51) x r r r. (5) x y and the formula (49) takes the form 1 ( x, ) 1. (53) ( xy,, ) r As a result, we get the following oerdetermined system of 1 equations (3) - (38) (41) * - (43), (46), (5), (51), (53), plus any of the equations from (47) and of unknowns,, u,, α, P, T, s, w,,, r. 8

9 5. Three-dimensional general stream We derie a system of equations describing the unsteady flow around the fixed rigid body in three-dimensional stream in the general case. Let S S11 S, S y, S3, S4 ux, S5 uy, S6 u, S7 x, S8 y, S9, S1 P, 1 x, S1 1 x, S13 x, S14 y, S15, S16, S17 T, S18 s, S19, S q, S qy, S3 q. (54) w, Then the system of 4 partial differential equations of the first order (3) - (38) (41) - (43), (46), (5), (51), (53), plus any of the equations from (47) can be written as H k S S S,,..., 1...3, k (55) r t We now turn to the coordinate system τ1, τ, n at point M (see Fig. 1). Then equations (55) can be written as S S S S Hk S,,,,..., 1...3, k (56) 1 n t Differentiating equations (56) in the direction n times, we obtain 55 equations of the form H Denote k i S S S S S,,,... 1 n t j j j n, 1...3, k 1...4, i.... (57) S S n, (58) where j...3, O h S S. Then 55 equations (57) can be written as: j j j j S S S S,,,..., 1...3, h , j...3. (59) 1 t We extend the ariable S S along the boundary of the body as follows: 1 1 α n α n. (6) Let g ( r be the equation of the solid surface in the coordinate system ( r (45). Similarly as in section 3, in the coordinate system ( r, the solid is static. Similarl dependence on deriatie to n in the transformed system of differential equations (59) is missing, and we obtain a closed system of 55 surface differential equations strictly along the boundary of a solid and the same number of ariables (58). In fact, some 9

10 ariables were already defined by (6) and (6), as well as by conditions of the heat transfer on surface of the body. Conclusion In this paper, using a special transformation of ariables we hae reduced the full set of hydrodynamic equations describing the unsteady flow around a rigid body in terms of threedimensional flow to a system of equations on the surface. These equations can greatly simplify the numerical simulation and explore the deeper features of the flow process. First, they reduce the dimension of the task by a unit; there is no need to sole the hydrodynamic equations in the boundary layer. Second, besides the gas elocity at the surface they allow defining how all other parameters characteriing the flow (such as, u, P, etc.) change at the boundary. It should be noted that not all of the obtained equations are with respect to time distinct systems. Therefore, in order to determine the stress distribution in addition to the initial data one should require examining boundary conditions. Through the preious, the information on the external flow is undoubtedly affecting the eolution of the whole effect of the flow on the body. They may need to be defined numerically using the additional code in space, the accuracy of which is important, but not in the entire olume, and only near the cures on the surface of the body along which the boundary conditions are determined. The systems of equations obtained in this paper are essentially three-dimensional ones, ie, degenerate into the two-dimensional case. In the two-dimensional case (see Fig. ) in order to oerride the hydrodynamic equations, it is necessary instead of transformations (15), (45) to use a different transformation similar to (A.1) (see Appendix A): dr dt α u α α r r( r, r r (,) r t, r r. (84), An additional condition should be implemented instead of (16) α α α α t. (85) Parameter is determined so that the boundary of the body did not moe in the coordinates r (see (3) and (6)). Despite the complexity, our description can successfully be used in applications, such as less expensie calculation of lift or drag coefficient in the external wing of impingement (including supersonic) flow, as well as the walls of the heat exchangers of nuclear reactors. The 1

11 Approach deeloped here is quite general and can be applied to other problems [14, 17]. If a more simple way is subsequently proposed to oerride the Naier-Stokes equations, the number of computations significantly reduced. This article may be of interest to researchers engaged in finding the analytical solutions of the equations of hydrodynamics. For example, according to the proposed method the hydrodynamic equations in the olume can be reduced to a system of equations on the surface and een get an oerdetermined system of surface equations. Consequentl it can already be reduced to an oerdetermined system of equations along a cure on the surface, etc. up to the analytical solution. It should be noted that the oerride method itself is quite simple and already described by the authors [14] where a method of oerriding the hydrodynamic equations described in Sec. and Sec. 4 is proided. Howeer, in this paper we apply it to a specific practical problem. Appendix A Integrals of motion [1] Consider the Euler equations of an ideal incompressible fluid in the 3D case, in the form u, t (А.1) diu. (А.) We also consider the Lagrangian ariables (i.e. markup) r r ( r, ). Multiply both sides of (A.1) to x. Then t x x di t u dix u diu x u x x u x u x u x. t Here we hae used the property of Lagrange ariables: x u t Consequentl t x u x u x x y. x y d dt x u x x x x, y x,, 11

12 / - time deriatie in the Lagrangian ariables, x, y, ) - where d dt / t u initial orticity distribution. Similarly y x, y,. x From the continuity equation (A.) one follows diu 1 d, dt u x, x, x, u y, x, x, u x,, y, ( and ( x, ) т.е. 1. (А.3) ( x, y, ) Expressing using (A.3), we also hae x, y, t x, x, y, t y x,, x, y, t, где x, y,, (see [1] page 31). We also consider the general equation of continuity y, u diu. (А.4) t Passing to the Lagrangian ariables, we find [6] i.e. d u x, dt x, d d, dt dt x, u y, x, x, u x,, ( x, ) ( x, ) - integral of motion. ( x, y, ) In the case and density x const, but s const, there is a clear connection between the pressure P S P the corresponding integrals are, y Similarl we obtain r,t r, и. The same results can be obtained if a more general transformation of the ariables is used instead of Lagrangian ariables 1

13 dr u( r, t) ( t) ( r, t), (А. 5) dt where r r( r и r r r, ), r r t - some function on t. Similar arguments ( t t, а can be established that in these ariables, x, y, and the Jacobian of this transformation do not depend explicitly on time in an ideal fluid. Instead of t we can take x,, t, including x,, t x,, t to satisfy di Expressing also, we obtain r,, r,, r,. In the case and density x t x y t y u. t const, but s const, there is a clear connection between the pressure P it is possible for the same reason in (A.5) to take x,, t P S, including. Then the corresponding integrals are x x r, ( r ) y y r, ( r ) r ( r ) ( r ). (А.6) и In fact, it requires only a negligible term P / in equation (A.1), taking into account the compressibility of the fluid, which is performed at a characteristic elocity of its motion being much less than the speed of sound. [1]. Going further. We sole the equation for the ector F F. (А.7) Consequentl Then F. и F. (А.8) The solution of this equation has the form F. (А.9) We now change the ariables dr u dt and thus implement the conditions r r( r, r r (,) r t, r r (А.1), t 13

14 и di. (А.11) Then these ariables x, y, and the Jacobian of this transformation do not depend explicitly on time. Similarl one can take into account compressibility. In the two-dimensional case, the change of ariables (A.1) has the form dr u, r r( r, r r (,) r t, r r, (А.1) t dt where 1 y. x Condition (A.11) becomes x y y x or G, t. (А.13) As a result, the change of ariables (A.1) can be written as dr u, t, r r( r, dt r r. t (А.14) These ariables allow ( x, y) x, y and 1. ( xy, ) 14

15 REFERENCES 1. L. D. Landau and E. M. Lifshit, Course of Theoretical Physics, Vol. 6: Fluid Mechanics (Nauka, Moscow,1986; Butterworth Heinemann, Oxford, 1987).. Williams F.A. Combustion Theory. - Benjamin Cummings, Menlo Park, CA, Ya. B. Zel doich, G. I. Barenblatt, V. B. Libroich, and G. M. Makhilade, The Mathematical Theory of Combustion and Explosion (Nauka, Moscow, 198; Consultants Bureau, New York, 1985). 4. A. N. Tikhono and A. A. Samarskii, Equations of Mathematical Physics (Nauka, Moscow, 1966; Doer, New York, 199). 5. A. A. Samarskii and Yu. P. Popo, Difference Methods for Soling Problems of Gas Dynamics (Nauka, Moscow, 198) [in Russian]. 6. M.L. Zaytse, V.B. Akkerman, A Nonlinear Theory for the Motion of Hydrodynamic Discontinuity Surfaces, JETP, 18, No 4, pp (9) 7. M.L. Zaytse and V.B. Akkerman, Method for Describing the Steady-State Reaction Front in a Two-Dimensional Flow, JETP Letters, 9, No 11, pp (1) 8. Bychko V., Zaytse M. and Akkerman V. Coordinate-free description of corrugated flames with realistic gas expansion // Phys. Re. E V P Joulin G., El-Rabii H. and K. Kaako K. On-shell description of unsteady flames // J. Fluid Mech V P Ott E. Nonlinear eolution of the Rayleigh-Taylor instability of thin layor // Phys. Re. Lett Vol P Book D.L, Ott E., Sulton A.L. Rayleigh-Taylor instability in the "shallow-water" approximation // Phys. Fluids Vol.17. N4. - P Зайцев M. Л., Аккерман В. Б. Метод снижения размерности в задачах арогидродинамики // Труды 5-ой научной конференции МФТИ "Современные проблемы фундаментальных и прикладных наук" Часть VI. - С Зайцев M. Л., Аккерман В. Б. Метод снижения размерности в переопределенных системах дифференциальных уравнений в частных производных // Труды 53-ой научной конференции МФТИ "Современные проблемы фундаментальных и прикладных наук" Часть VII, Т С V.B. Akkerman and M.L. Zaytse, Dimension Reduction in Fluid Dynamics Equations, Computational Mathematics and Mathematical Physics, Vol. 8 No. 51, pp (11) 15

16 15. G. V. Korene, Tensor Calculus (Moscow Institute of Physics and Technolog Moscow, 1996) [in Russian]. 16. J. A. Schouten, Tensor Analysis for Physicists (Clarendon, Oxford, 1954; Nauka, Moscow, 1965). 17. M.L. Zaytse, V.B. Akkerman, Free Surface and Flow Problem for a Viscous Liquid, JETP, 113, No 4, pp (11) 16

17 FIGURES Fig. 1. Rigid body in a iscous medium 17

18 Fig.. A solid in iscous fluid in the two-dimensional case 18

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