On the Evaluation of the MD Passive WDF Discretization Method for a Flow through Cylinder

Transcription

1 On the Evaluation of the MD Passive WDF Discretization Method for a Flow through Cylinder F. N. KOUMBOULIS * M. G. SKARPETIS * and B. G. MERTZIOS ** * University of Thessaly School of Technological Sciences Dep. of Mechanical & Industrial Eng. 8 Pedion Areos Volos Greece. ** Democritus University of Thrace Dep. of Electrical and Computer Eng. 7 Xanthi Greece. Fax: --97 or 7 Abstract: In this paper a complete algorithm is developed to evaluate the multidimensional (MD) wave digital filter (WDF) method for the numerical integration of nonlinear (NL) partial differential equations (PDEs) and particularly those referring to the solution of the Euler equations. The evaluation is accomplished for the case of flow around a cylinder. The resulting discrete model is compared to the respective analytic solutions via simulations results. The advantages of the method at hand namely robustness and passive parallelism are verified. KeyWords:- Wave digital filters Discrete Modeling in Transfer phenomena Nonlinear dynamic Infinite dimensional systems Fluid Mechanics. Introduction The Euler equations describe the dynamics of an inviscid fluid thus having a variety of applications in fluid mechanics particularly in industrial processes. The Euler equations is a set of NL PDEs for which a variety of numerical methods have been developed. Here we focus on the method in [-] where the numerical integration of the Euler equations is accomplished via a generalized Kirchhoff circuit realization and after discretization via WDF principles. The method in [- 9] claims in establishing an algorithm being robust to data variations and having massive parallelism. For validation purposes the case of inviscid and incompressible flow through a cylinder is studied. This particular case has a simple and elegant analytic solution being easily comparable to any other numerical result. Particularly a complete algorithm simulating the MD WDF equations for the case of the flow around a cylinder of radius R is implemented in MATHEMATICA. In the resulting stable discrete model the web of incident waves facilitates the computation of the waves at different points and so the derivation of the velocities and pressure of the fluid at these points. The computation is accomplished by solving recursively a corresponding set of nonlinear algebraic (not dynamic) equations. The numerical computation of velocities and pressure of the fluid has been completed at any point out of a cylinder with radius grater than or equal to R (area of passivity). On the basis of this algorithm simulation results are derived for several areas around the cylinder. It is important to mention that the simulation results are satisfactory close to those derived by the analytic solution of the application at hand.. Flow through a Cylinder.. D Euler Equations A D steady inviscid and incompressible flow is expressed by the following two PDEs [8]: Øq =p + q $ =q = (.) Øt = $q = The vector t = [ t t ] T is the vector of the two spatial variables x and y (t = y t = x). Based on this definition the nabla operator = is expressed equivalently as follows: = =. The Ø T Ø Øt Øt velocity vector is denoted by q = q q T where q is the velocity of the fluid along the direction t and q is the velocity of the fluid along the direction t is the density (constant) and p is the pressure of the fluid in the control volume.. Analytic solution of the pressure and velocities Consider inviscid incompressible flow through a cylinder of radius R. Let U denotes the free stream velocity then the velocity of the fluid q = q q T and the pressure p at a particular point with coordinates x y T = t t T is computed to be [8]: q (x y) = Uy R UR (x +y ) x +y + U q (x y) = UxyR (x +y ) p(x y) = U R (R x +y ) (x +y ) or equiv. in terms of the spatial variables and t t (.a) (.b) q (t t ) = Ut R (t +t UR ) t +t + U q (t t ) = UttR (t +t ) p(t t ) = U R (R t +t ) (t +t ) (.a) (.b)

2 .Transformation of Euler equations In this section three equivalent expressions of the Euler equations (.) will be presented. These equivalent expressions will facilitate the determination of the passive generalized Kirchhoff circuit realization that corresponds to the particular Euler equations and will be presented in Section. It is mentioned that the steps required for the derivation of the equivalent forms have first been presented in [ 9]. First using the notation D = [D D ] T D k = Ø (k = ) equation (.) can be rewritten as Øt k qk D k q + =p = Dk (q k ) = (.) k= k=. Normalization The goal of this transformation is to introduce new arbitrary parameters in the Euler equations. These parameters will be useful in proving passivity. Define the following normalized quantities qˆ = qˆ qˆ T = (.) q a q ˆ = p pˆ = a p p where p are positive constants and a q are arbitrary constants.using (.) and the normalization identity w D z w = wdz + zdw (holding true for every real functions w z with w scalar ( w m ) and z vector) eq. (.) takes on the form ˆa q kq D k = j = k= qˆj ˆa q kq + D j pˆ (.a) Dk k= ˆ qk q = (.b). Hadamard Transformation To obtain a MD passive circuit realization for the equations in (.) the following Hadamard transformation is proposed to be applied ([] and []) t = H t where t = t t T and where H is the orthogonal matrix [] H = H T =. Defining the new derivative operators D k = Ø Øt k (k = ) and using these definitions the following expression is derived D = H T D. Clearly the operators D i may be expressed in terms of the new derivative operators: D = ( D + D ) D = (D or inversely + D ) D = ( D + D ) and D = ( D. Using the above + D ) transformations and after some manipulations equations (.) may be written as: u + u + u + u + u + u = (.a) u + u + u + u + u + u = (.b) u + u + u + u + u + u = (.c) where the variables u ji are related to the normalized velocities as in the following relations u = L D ( L qˆ) u = L D ( L qˆ) u = L D ( L qˆ) u = L D ( L qˆ) (.a) u = L D (qˆ) u = L D (qˆ) u = L D (qˆ) u = L D (qˆ) (.b) u = D (qˆ pˆ) u = D (qˆ + pˆ) u = D (qˆ + pˆ) u = D (qˆ + pˆ) (.c) u = L D ( L pˆ) u = L D ( L pˆ) (.d) u = u u = u u = u u = u (.e) The variables u ij being a nonlinear transformation of the fluid velocities are the unknowns in the nonlinear system of equations (.). The parameters L i and L i are given by L = ˆa ( q+q+q)q L = ˆa (q+q+q)q L = L = ˆa q (.a) L = ˆ ( q + q + q ) + K pˆ a q pˆ L = ˆ (q + q + q ) + K pˆ a q pˆ (.b) where K are arbitrary real constants with m = p dp = p dp (.7) ˆ u p where for the derivation of the above relations it has been assumed that the fluid is barotropic i.e. that the pressure is a unique function of the density. For incompressible fluid (i.e. = const ) the parameter p may be chosen as follows p =. Thus according to (.) we have ˆ = and consequently since dp = (p + K ) where K is the integration arbitrary constant the relation (.7) takes on the form ˆ = = + K p.. Generalized Circuit Realization It is observed that the transformed form of the Euler equations namely the three equations in (.) describe a generalized Kirchhoff circuit [ 9] involving three loops appropriately interconnected.. Main Loop The generalized Kirchhoff circuit involving the three loops is shown in Figure. q q L D' L D' L D' L D' D' D' L D' L D' L D' L D' D' D' -/ / p Figure : Generalized Kirchhoff Circuit p L D' L D'

3 where the symbol L i (i = ) D i (i = ) denotes the operator applied to the current qˆj (j = ) to yield the voltage u ji = L i D i ( L i qˆj) (see (.a)). An analogous definition holds for L i D i (i = ). For more details see relation (.). The circuit in Figure is passive if every element of the circuit is positive i.e. if L m L m L m L m L m L m (.) According to the definitions in (.) the choice K = and the relations ˆ = = + K p the conditions in (.) can be rewritten as follows L = a ( q+q+q)q m L = a (q+q+q)q m L = L = a q (.a) m L = + K m L = + K m (.b) pˆ pˆ Using the third of the relations in (.) the conditions in (.b) take on the form L = + K ap m L (.) = + K ap m In concluding to certify the passivity of the circuit in Figure it suffices to find appropriate arbitrary parameters a q and K such that the passivity conditions in (.a ) are satisfied. To this end start by defining p max > and q min > to be two positive real numbers being the upper and the lower bound of p(t) and q (t) q (t) respectively i.e. p max m p(t) and q min [ q (t) q (t) t c ß where ß is the set of all points of the t plane out of the circle of radius R ( t + t m R ). These two positive numbers will be proven to exist and they will be determined in to the following two lemmata. Lemma.: It holds that p(t) [ U t c ß (.) Proof: From relation (.b) it is observed that p(t t ) = U R (R t + t ) [ (t + t ) U R (t + t ) R + t + t = U R (t + t ) + U R t + t Since t + t m R the inequality (.) is derived. According to Lemma. select p max = U. Lemma.: It holds that q (t) q (t) m U / t c ß (.) Proof: From relation (.a) and since q (t) m t c ß it is readily observed that q (t) q (t) = Ut R (t + U +t ) UR U t t R t +t (t = +t ) U R (t t )+U (t +t ) U t t R (t +t ) Since t + t m R > and U > to prove (.) it suffices to prove the following inequality R (t t ) + (t + t ) t t R m R (.) To prove (.) t c ß it suffices to prove that (.) holds on a circle of radius R i.e. for t + t = R m R. For this case express the variables t and t in the parameterized form t = R sin and t = R cos to yield R R [sin( )] [cos( )] + R sin( ) cos( ) R R m (.7) Since R m R it suffices to prove that [sin( )] [cos( )] + + sin( ) cos( ) m or equivalently that [sin( )] sin( ) cos( ) + m (.8) For the inequality to be satisfied it is necessary for the polynomial to have complex roots or one root of multiplicity i.e. cos( ) [. The latter inequality is clearly true. On the basis of Lemmata. and since a > the inequalities in (.a) and (.) can be satisfied if a U q (.9) m a q a q m K m au Theorem.: The generalized Kirchhoff circuit in Figure is passive if the arbitrary parameters q and K are chosen in terms of the arbitrary parameters a > and to be q = au = (.) au + K = au. Discretization Algorithm In order to transform the circuit of Figure to an equivalent reference WDF circuit the circuit must first be discretized. To this end the trapezoidal rule must be applied (see f.e. [7]) to the inductances occurring in the circuit. Consider the voltage u(t ) defined by the following generalized inductance description u = L(t ) D( L(t ) i(t )) (.) where D = Ø($) + Ø($) and i(t is a generalized Øt Øt ) current. Apply the general trapezoidal rule to the relation (.) to yield u L ( t ) + u L ( t T ) = (/T ) ( L i)(t ) ( L i)(t T ) (.) where T is a vector of shift of the sampling T = [T T ] T = T T > ; = [ ] T i m (.) The above sampling vector can be interpreted as follows: T is the basic sampling period (scalar and constant) T = T T T is the vector of sampling period for each transformed space and time coordinate. The coefficients i are the weight coefficients for every sampling period. Clearly as i d the discretization becomes perfectly accurate. It is noted that in many cases since all elements of

4 t = t t T are in spatial dimensions the weighting coefficients can be considered to be equal i = T i = T = T The trapezoidal rule can be expressed in terms of an operator let (T ) r $ as follows u(t ) = (T ) r(t )i(t ) ; r(t ) = L(t ) (.) Define T = T T = T. Based upon the above discretization rule the discretized circuit corresponding to the passive circuit of Figure is derived to be as in Figure that follows. q q Ä(T'){.} Ä(T' ){. } Ä(T'){.} Ä(T' ){.} Figure : Reference circuit T -/ p Ä(Ô'){r (t') } Ä(Ô'){r (t') }. WDF Realization The analytic and circuit forms of the WDF realization of the reference circuit in Figure can easily be derived on the basis of the results in [9]. The analytic form of the WDF realization is the formulation of the numerical algorithm integrating the Euler equations. In order to derive a robust algorithm it is suitable to adopt power waves instead of the voltage waves usually preferred in wave digital filtering []. Thus for a port of voltage u current i and nonconstant port resistance R m the forward a and the backward wave b are defined as follows b = u+ri a = u Ri. R R The reference circuit in Figure can be analyzed into three loops where the only the third loop is connected to the other three. Based upon this observation and the theory of WDF [] the reference circuit can be realized by three WDF circuits where the third is connected to the rest two by appropriate adaprors. Based on this WDF realization the computational point of view the problem of solving the Euler equations can be reduced to that of solving a nonlinear algebraic system of equations. 7. Simulation Results The velocity and the pressure of the fluid are computed into an area let. The area without loss of generality is considered to be a parallelogram The parallelogram is defined in the plane [t t ] T and so it corresponds to a parallelogram in the original / [t t ] T plane after appropriate rotations. The rotation matrix is the Hadamard transformation. The boundary conditions are considered to be a profile of the velocities and the pressure along the two sides of the parallelogram let OA and OB. Furthermore the first derivatives of the velocities and the pressure are considered to be known along the two sides. In particular the following quantities are considered to be known on the sides OA and OB q (t t ) q (t t ) Øq(t t ) t ) t ) t ) p(t t ) Øt Øq(t Øt Øq(t Øt Øq(t Øt Øp(t t ) Øp(t t ) Øt Øt (7.) To implement an algorithm numerically integrating the D Euler PDEs in (.) the sampling period T is considered to be equal to.. The web of the area points are defined as follows: n = t t T n = t t T (7.) where [[$]] denotes the integer part of the argument real number. The algorithm is based upon the computations of the incident waves (backward and forward) of the points [n n ] of the sides OA and OB. Then the incident waves are computed at all the sampling points namely through the web. In particular the forward waves a(n + n + ) can be computed by the waves at the points a(n + n ) and a(n n + ) After the computation of the incident waves over the web of sampling points the velocities q (n n ) q (n n ) and the pressure p(n n ) are computed by solving the nonlinear algebraic equations (.8) with respect to q q and p. The nonlinear equations are solved by using the Newton - Raphson method. To reduce the sensitivity of this method let say at the point (n n ) the initial point of the method is chosen to be the solution of the nonlinear equation at the point (n n ). The iterations of the method are chosen to be 8 while the numerical precision is chosen to be digits. The results of the simulation are illustrated via D plots of the velocities and pressure for both the analytic and numerical solutions (see Figures -8). To derive the simulation results a software program has been constructed in MATHEMATICA. 8. Conclusions In this paper a complete algorithm has been developed to evaluate the multidimensional (MD) wave digital filter (WDF) method for the numerical integration of nonlinear (NL) partial differential equations (PDEs) and particularly those referring to the solution of particular Euler equations. The evaluation has been accomplished via simulation results which are compared to the respective analytic solutions for the case of flow around a cylinder.

5 References [] A. Fettweis Discrete modeling of losseless fluid dynamic systems AE Ü vol. pp [] R. Bernhardt and D. Dahlhaus Numerical Integration of the Euler equations by means of Wave Digital Filters ICASSP Australia 99. [] A. Fettweis Wave digital filters: Theory and Practice Proc. IEEE vol. 7 pp [] A. Fettweis On assessing robustness of recursive digital filters European Transactions on Telecommunications vol. pp [] A. Fettweis Multidimensional wave digital filters for discrete-time modeling of Maxwell s equations Int. J. Num. Modeling vol. pp [] A. Fettweis and G. Nitsche Numerical integration of partial differential equations using principles of multidimensional wave digital filters Journal of VLSI Signal Proc. vol. 99 pp. 7-. [7] A. Fettweis and G. Nitsche Numerical integration of partial differential equations by means of multidimensional wave digital filters Proc. IEEE Int. Symp. Circuits and Systems vol. New Orleans LA. May 99 pp [8] F. M. Whitaker Introduction to Fluid Mechanics Prentice-Hall Enngelwood Cliffs N.J. 98. [9] F. N. Koumboulis M. G. Skarpetis and B. G. Mertzios On the Derivation of the Nonlinear Discrete Equations Numerically Integrating the Euler PDEs Discrete Dynamics in Nature and Society vol. pp Figure : Original velocity q (n n ) of the fluid for n c {... } n c {... } Figure : Computed velocity q (n n ) of the fluid for n c {... } n c {... }... Figure : Original velocity q (n n ) of the fluid for n c {... } n c {... } Figure 7: Original pressure p(n n ) of the fluid for n c {... } n c {... }.... Figure : Computed velocity q (n n ) of the fluid for n c {... } n c {... } Figure 8: Computed pressure p(n n ) of the fluid forn c {... } n c {... }