Integral transform solution for the lid‐driven cavity flow problem in streamfunction‐only formulation

Abstract

The basic ideas in the generalized integral transform technique are further advanced to allow for the hybrid numerical-analytical solution of the two-dimensional steady Navier-Stokes equations in streamfunctiononly formulation. The classical lid-driven square cavity problem is selected for illustration of the approach. The corresponding biharmonic-type non-linear partial differential equation for the streamfunction is integral transformed in one of the co-ordinates and an infinite system of coupled non-linear ODES for the transformed potential results in the other independent variable. Upon truncation to an appropriate finite order, the ODE system is numerically solved by well-established algorithms with automatic error control devices. The convergence behaviour of the eigenfunction expansion is demonstrated and reference results are provided for typical values of Reynolds number.

Verification and validation technologies for gas dynamic simulations

“…Верификацию также проводят с применением высокоточных решений системы дифференциальных уравнений в частных производных для определенных граничных условий. Например, используют решения, полученные для случаев: несжимаемого ламинарного течения над полубесконечной плоской пластиной [70,74,78], несжимаемого ламинарного течения жидкости в квадратной каверне с движущейся крышкой [149][150][151][152][153], несжимаемого ламинарного течения около круглого цилиндра бесконечной длины [44,45,57,154]. Очевидно, что гарантируемая точность эталонных решений снижается при переходе от обыкновенных дифференциальных уравнений к уравнениям в частных производных.…”

Two-dimensional generalized finite integral transform method for new analytic bending solutions of orthotropic rectangular thin foundation plates

“…Besides the well-known semi-inverse superposition method [12] that was applied for some simple plate problems, few new analytic methods have been found in the literature, including the symplectic approach [13][14][15][16], Fourier-type finite integral transform method [17,18], etc. It is notable that the one-dimensional generalized finite integral transform method has been applied in the fields of thermodynamics and fluid mechanics [19,20], by which solving PDEs reduces to solving ordinary differential equations where special mathematical techniques are still required. This paper presents a first endeavor to extend the one-dimensional generalized finite integral transform to two-dimensional transform for new analytic bending solutions of orthotropic rectangular thin foundation plates, with focus on typical clamped plates that were difficult to solve by the other analytic methods.…”

“…are flexural rigidities in the x and y directions, respectively; is the deflection,   , q x y the load, and K the Winkler foundation modulus.The following two-dimensional generalized finite integral transform pair is defined:Yy are the vibrating beam functions[20]:…”

A review of hybrid integral transform solutions in fluid flow problems with heat or mass transfer and under Navier–Stokes equations formulation

“…In a relatively short period of time it was extended to different classes of problems, including nonlinear diffusion and convection-diffusion [16][17][18][19] and irregular domains in parabolic and elliptic formulations [20][21][22][23]. It would not take long for the GITT to be challenged by the solution of fluid flow problems governed either by the boundary layer equations or the Navier-Stokes equations [24,25]. Since then, the hybrid method was progressively extended and new classes of problems and applications have been dealt with, and it has been reviewed at different stages and sources [26][27][28][29][30][31][32][33][34].…”

“…The integral transform analysis of fluid flow problems governed by the Navier-Stokes equations has required the proposition of new eigenfunction expansions, other than those normally employed in diffusion or convection-diffusion problems, directly derived from the general Sturm-Liouville eigenvalue problem. Along the years, in the present methodological context, the Navier-Stokes equations have been mostly dealt with in the streamfunction-only formulation [25,[35][36][37][38][39][40][41][42][43][44][45][46], and less frequently in the primitive variables formulation [47,48]. In two-dimensional problems, the streamfunction formulation offers the advantages of automatically satisfying the continuity equation and eliminating the pressure field.…”