THE LIE-GROUP SHOOTING METHOD FOR BOUNDARY LAYER EQUATIONS IN FLUID MECHANICS *

Transcription

1 Conference of Global Chinese Scholars on Hydrodynamics HE LIE-GROUP SHOOING MEHOD FOR BOUNDARY LAYER EQUAIONS IN FLUID MECHANICS * CHANG Chih-Wen CHANG Jiang-Ren LIU Chein-Shan Department of Systems Engineering and Naval Architecture National aiwan Ocean University Keelung China Department of Mechanical and Mechatronic EngineeringNational aiwan Ocean University Keelung China cjr@mailntouedutw ABSRAC: In this paper we propose a Lie-group shooting method to tackle two famous boundary layer equations in fluid mechanics namely the Falkner-Skan and the Blasius equations We can employ this method to find unknown initial conditions he pivotal point is based on the erection of a one-step Lie group element G() and the formation of a generalized mid-point Lie group element G(r) hen by imposing G() = G(r) we can seek the missing initial conditions through a minimum discrepancy of the target in terms of the weighting factor r (1) It is the first time that we can apply the Lie-group shooting method to solve the boundary layer equations Numerical examples are worked out to persuade that this novel approach has good efficiency and accuracy with a fast convergence speed by searching r with the minimum norm to fit two targets KEY WORDS: One-step group preserving scheme Falkner-Skan equation Blasius equation boundary value problem Lie-group shooting method estimation of missing initial condition 1 Introduction he Falkner-Skan equation arised in the study of laminar boundary layers exhibiting similarity he solutions of the one-dimensional third-order boundary-value problem depicted by the well-known Falkner-Skan equation are the similarity solutions of the two-dimensional incompressible laminar boundary layer equations his is a nonlinear two-point boundary value problem for which no closed-form solutions are available he first numerical treatment of the problem was presented by Hartree [1] he mathematical treatments of this problem by Weyl [] and Rosenhead [3] have principally concentrated on attaining results of existence and uniqueness Na [4] Cebeci and Keller [5] and Smith [6] have addressed other numerical methods for solving the problem All these approaches have mainly employed shooting or invariant imbedding Later those methods presented by Asaithambi [7-11] have improved the performance of the previous methods by reducing the amount of the computational effort When a two-dimensional (D) steady flow of an incompressible constant property fluid with very low viscosity and high Reynolds number moves promptly over a semi-infinite flat plate the friction between the fluid and the flat plate will induce the fluid to be obstructed within a thin region immediately adjacent to the boundary layer he governing equation describing the boundary layer with such fluid characteristics and boundary conditions is called the Blasius equation [1] In 198 Blasius [13] gave a solution in the form of a power series for the Blasius equation and since then it has led to much attention on solving this equation by developing different techniques öpfer [14] began to adopt the Runge-Kutta algorithms to solve this equation and until the age of Howarth [15] the numerical solution with the Runge-Kutta method is still not as accurate and reliable as presently shown in tabulated result [1] Apart from this Lock [1617] investtigated two cases the lower stream was at rest as well as in motion hereafter Liao [1819] proposed a systematic depiction of a new kind of analytic technique for non-linear problems namely the homotopy analysis method (HAM) He applied it to give an explicit and analytic solution of the D laminar viscous flow over a semi-infinite flat plate his method may have higher accuracy but it is very complex in expression In addition Yu and Chen [] converted the Blasius equation to a pair of initial value problems and then solved them by a differential transformation method o speed up the convergent rate and the accuracy of calculation the entire domain needs to be divided into sub-domains * Project supported by the National Science Council (Grant No: NSC 94-1-E-19-5) 13

2 Biography : CHANG Jiang-Ren (196-8) Male Professor In this paper we propose a Lie-group shooting method to tackle these two famous boundary layer equations in fluid mechanics Our approach is based on the group preserving scheme (GPS) developed by Liu [1] for the integration of initial value problems and originally its extension is only applied to the solution of second order boundary value problems by Liu [] It will be clear that our method can be applied to two famous boundary layer equations since we are able to search the missing initial condition through a minimum solution of r in a compact space of r (1) the factor r is used in a generalized mid-point rule for the Lie group of one-step GPS Especially the proposed scheme is easy to implement and time saving hrough this study we may have an easy-implementation and accurate Lie-group shooting method (LGSM) used in the calculations of two famous boundary layer equations One-Step GPS 1 he GPS Although we do not know previously the symmetry group of nonlinear differential equations system Liu [1] has embedded it into an augmented system and found an internal symmetry of the new system hat is for an ODEs system with dimensions n: n u = f ( u t) u R t R (1) we can embed it into the following n+1-dimensional augmented system: d dt f ( u t) n n d u u u X : = = ( ) dt u f u t u u () It is obvious that the first row in Eq () is the same as the original Eq (1) but the inclusion of the second row in Eq () gives us a Minkowskian structure of the augmented system for X satisfying the cone condition: X gx = u u u = (3) Where I 1 g = (4) 1 1 is a Minkowski metric I n is the identity matrix of order n and the superscript stands for the transpose he cone condition is a natural constraint imposed on the system () Consequently we have a n+1-dimensional augmented system: X = AX (5) with a constraint (3) f ( u t) n n u A : = (6) f ( u t) u is an element of the Lie algebra so(n1) satisfying A g + ga = (7) herefore Liu [1] has developed a grouppreserving numerical scheme as follows: X l + 1 = G( l ) Xl (8) Xl denotes the numerical value of X at the discrete time tl and G(l) SO o (n1) satisfies G gg = g (9) det G = 1 (1) G (11) > G is the th component of G Generalized mid-point rule Applying scheme (8) to Eq (5) with a specified initial condition u() = u we can compute the solution u(t) by GPS Assuming that the stepsize used in GPS is Δt = / K and starting from an initial augmented condition X = X( ) = ( u u ) we want to calculate the value X ( ) = ( u ( ) u( ) ) at a desired time t = By applying Eq (8) step-by-step we can obtain X = GK ( Δt) G 1 (Δt) X (1) X approximates the exact X() with a certain accuracy depending on Δt However let us recall that each G i i = 1 K is an element of the Lie group SO o (n1) and by the closure property of Lie group GK ( Δt) G1(Δt) is also a Lie group denoted by G Hence we have X = GX (13) his is a one-step transformation from X to X We can calculate G by a generalized mid-point rule which is obtained from an exponential mapping of A by taking the values of the argument variables of A at a generalized mid-point he Lie group generated form A so(n1) is known as a proper orthochronous Lorentz group which admits a closed-form representation as follows: 14

3 ( a 1) ˆ ˆˆ bf I + ff ˆ ˆ f f G = ˆ bf a ˆ f (14) uˆ = ru + (1 r ) u (15) f ˆ = f ( uˆ) (16) ˆ f a = cosh (17) uˆ ˆ f b = sinh (18) uˆ Here we employ the initial u and the final u through a suitable weighting factor r to calculate G < r <1 is a parameter he above method is applied a generalized mid-point rule on the calculation of G and the result is a single-parameter Lie group element G(r) 3 A Lie group mapping between two points on the cone Let us define a new vector fˆ F : = (19) uˆ such that Eqs (14) (17) and (18) can also be expressed as ( a 1) bf I + FF F F G = () bf a F a = cosh( F ) (1) b= sinh( F ) () uf From Eqs (13) and () it follows that = u + ηf (3) F u u = a u + b F (4) ( a 1) F u + b u F η : = (5) F Substituting 1 F = ( u u ) (6) η into Eq (4) we obtain u ( u u) u = a + b (7) u u u u u u a = cosh (8) η u u b = sinh (9) η are obtained by inserting Eq (6) for F into Eqs (1) and () Let [ u u] u : = (3) u u u S u u (31) : = and from Eqs (7)-(9) it follows that u S S = cosh + sinh (3) u η η By defining S Z : = exp (33) η we obtain a quadratic equation for Z from Eq (3): u (1 + ) Z Z + 1 = (34) u he solution is found to be u f u f + 1+ cos θ u u Z = (35) 1+ and then from Eqs (33) and (31) we obtain u u η = (36) lnz hus between any two points u ) and ( ( u u u ) on the cone there exists a Lie group element G SO o (n1) mapping u ) onto ( u u ) which is given by ( u u u = G (37) u u G is uniquely determined by u and u through Eqs ()-() (6) and (36) 15

4 3 he LGSM for Falkner-Skan and Blasius Equations Let us consider the Falkner-Skan equation: f + f f + υ (1 f ) = (38) subject to the boundary conditions f ( ) = f () = f ( ) = 1 (39) the prime stands for the differential with respect to ξ Note that problem (38) and (39) is depicted on a semi-infinite physical domain Because this is not very convenient for computations it is used to replace ξ = in condition (39) with the sufficiently large ξ [7-11] Let y 1 = f y = f and y3 = f We can rewrite Eqs (38) and (39) y 1 = y (4) y = y 3 (41) y 3 = y1y 3 υ ( 1 y ) = Y( x y1 y y3) (4) y 1( ) = α y1( ) = A (43) y ( ) = β y( ) = B (44) y 3( ) = δ y3( ) = C (45) A and δ are three unknown constants and α = β = B = 1 and C = are four given constants Let y1 u : = y y3 (46) From Eqs (3) (43)(44) and (45) it follows that F1 A α 1 F : = F = B β η F3 C δ (47) Starting from an initial guess of (A δ) we use the following equation to calculate η: ( α A) + ( β B) + ( δ C) η = lnz (48) in which Z is calculated by A + B + C A + B + C + (1 cos θ) α + β + δ α + β + δ Z = 1+ (49) α( A α) + β( B β) + δ( C δ) = ( α A) + ( β B) + ( δ C) α + β + δ (5) he above three equations were obtained from Eqs (36) (35) and (3) by inserting Eq (46) for u When comparing Eq (47) with Eq (19) and with the aid of Eqs (15) (16) and (4)-(45) we obtain η[ A = α + (51) η[ rδ + (1 r ) C] B = β + (5) η{[ rα + (1 r ) A][ rδ + (1 r ) C] C = δ + υ{1 [ }} + (53) hrough some calculations we obtain E + E 4DF = Dη (54) η[ A = α + (55) ( B β ) δ = rη (56) υ{1 [ } D : = (57) E : = (B β )[ rα + (1 r ) A] (58) (B β ) F : = r (59) Ŷ : = Y( r rα + (1 r ) A + (1 r ) B rδ + (1 r ) C) (6) : = [ rα + (1 r) A] + [ + (1 r) B] + [ rδ + (1 r) C] (61) ( α A) + ( β B) + ( δ C) η : = lnz (6) When ν of Eq (4) is equal to Eq (4) is the Blasius equation herefore we can rewrite Eq (54) r[ rα + (1 r ) A] η (63) he above derivation of the governing equations (48)-(63) is stemmed from by letting the two F in Eqs (19) and (6) be equal which is essentially identical to the specification of G() = G(r) in terms of the Lie group elements G() and G(r) For a specified r Eqs (54) (55) (56) and (63) can be used to generate the new (A δ) by repeating the above process in Eqs (48)-(63) until (A δ) converges according to a given stopping criterion with ε = :

5 ( Ai + 1 Ai ) + ( i + 1 i ) + ( δi + 1 δi ) ε1 (64) If δ is available we can return to Eqs (4)-(44) but with merely integrating the following equations by a forward integration scheme as the one given in Section : y 1 = y (65) y = y 3 (66) y 3 = y1y 3 υ ( 1 y ) = Y( x y1 y y3) (67) y 1() = α (68) y () = β (69) y 3() = δ (7) So far we have not yet said that how to determine r r Let yn () denote the above solution of yn at We start from r = 1/ to determine δ by Eqs (48)-(64) and then numerically integrate Eqs (65)-(7) from t = to t = and compare the end value of y () with the exact B hen we apply the minimum norm to fit the two targets of y () = 1 and y 3 () = which requires us to calculate Eqs (65)-(7) at each of the calculation r r of ( y B) + ( y3 C) until it is small enough to satisfy the criterion of r r ( y B) + ( y3 C) < εmin ε min = 36 is a given error tolerance Because the numerical method is very stable we can fast carry off the correct range of r through some trials and modifications 4 Interpretative Results Following Section 3 when the factor ν is equal to 5 1 and we apply the shooting method to the Falkner-Skan equation with an initial (A δ) = (3 ) and through some trials we take r = [ ] r = [ ] and r = [ ] respectively By using a stepsize Δ ξ = 1 the numerical results are shown in able 1 he solutions corresponding to ν > have become known as accelerating flows and those corresponding to ν = are called constant flows From able 1 it is apparent that our results are in great agreement with those reported previously in the literature When the factor ν is equal to we apply the shooting method to the Blasius equation with an initial (A δ) = (3 ) and through some trials we take r = [ ] By using a stepsize Δ ξ = 1 the numerical results are shown in able From able it is obvious that our results are in excellent agreement with those given by Cortell [3] able 1 Comparison of computed δ for 5 ν ν LGSM Asaithambi [8] Asaithambi [9] Asaithambi [111] f f and f f of f of able Values of functions ξ f of f of f of f of LGSM C LGSM C LGSM C LGSM-Lie-group shooting method; C-Cortell s solution [3] 5 Conclusions here are two significant points deserved further inform in this paper he first is the erection of a one-step group G() and the full use of Eqs (3) and (4) which are the Lie group transformation between initial conditions and final conditions in the augmented Minkowski space hen another one is the use of a generalized mid-point rule to erect another Lie group element G(r) In order to evaluate the missing initial conditions for boundary value problems we have employed the equation G() = G(r) to derive algebraic equations Hence we can solve them through a minimum solution in a compact space of r (1) Numerical examples of the Falkner-Skan and the Blasius equations were examined to ensure that the new algorithm has a fast convergence speed on the solution of r in a pre-selected range smaller than ( 1) by using the minimum norm to fit two targets which usually required only small number of iterations hrough this paper it can be concluded that the new shooting method is accurate effective and stable Its numerical implementation is very simple and the computation speed is very fast hus it is highly advocated to be used in the numerical computations of two famous boundary layer equations in fluid mechanics References [1] HARREE D R On an equation occurring in Falkner and Skan s approximate treatment of the equations of the boundary layer [J] Proceedings of the Cambridge Philosophical Society : 3-39 [] WEYL H On the differential equations of the simplest boundary-layer problem [J] Annals of Mathematics : [3] ROSENHEAD L Laminar boundary layer [M] Oxford: Clarendon Press 1963 [4] NA Y Computational methods in engineering boundary value problems [M] New York: Academic Press 1979 [5] CEBECI ; KELLER H B Shooting and parallel 17

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