Numerical simulations of inviscid and viscous free‐surface flows with application to nonlinear water waves

The purpose of the present study is to establish a numerical model appropriate for solving inviscid/viscous free‐surface flows related to nonlinear water wave propagation. The viscous model presented herein is based on the Navier–Stokes equations, and the free‐surface is calculated through an arbitrary Lagrangian–Eulerian streamfunction‐vorticity formulation. The streamfunction field is governed by the Poisson equation, and the vorticity is obtained on the basis of the vorticity transport equation. For computing the inviscid flow the Laplace streamfunction equation is used. These equations together with the respective (appropriate) fully nonlinear free‐surface boundary conditions are solved using a finite difference method. To demonstrate the model feasibility, in the present study we first simulate collision processes of two solitary waves of different amplitudes, and compute the phenomenon of overtaking of such solitary waves. The developed model is subsequently applied to calculate (both inviscid and the viscous) flow field, as induced by passing of a solitary wave over submerged rectangular structures and rigid ripple beds. Our study provides a reasonably good understanding of the behavior of (inviscid/viscous) free‐surface flows, within the framework of streamfunction‐vorticity formulation. The successful simulation of the above‐mentioned test cases seems to suggest that the arbitrary Lagrangian–Eulerian/streamfunction‐vorticity formulation is a potentially powerful approach, capable of effectively solving the fully nonlinear inviscid/viscous free‐surface flow interactions. Copyright © 2009 John Wiley & Sons, Ltd.     

Conservative calculations of non-isentropic transonic flows

The classical potential formulation of inviscid transonic flows is modified to account for non-isentropic effects. The density is determined in terms of the speed as well as the pressure, which in turn is calculated from a second-order mixed-type equation derived via differentiating the momentum equations. The present model differs in general from the exact inviscid Euler equations since the flow is assumed irrotational. On the other hand, since the shocks are not isentropic, they are weaker and are placed further upstream compared to the classical potential solution. Furthermore, the streamline leaving the aerofoil does not necessarily bisect the trailing edge. Results for the present conservative calculations are presented for non-lifting and lifting aerofoils at subsonic and transonic speeds and compared to potential and Euler solutions.     

Viscous transonic spiral and radial flow

Similarity solutions of the viscous transonic equation describing source and source vortex flows have been found. These solutions contain shock-like transitions from the supersonic to the subsonic branch of the corresponding inviscid solutions, while the singularity near the sonic point of the inviscid solutions is shifted to a smaller radius. It is shown that this similarity solution is identical to the transonic viscous compressible source and sink flow solutions of Wu (1955) and Sakurai (1958).     

Computation of unsteady transonic flow over a fighter wing using a zonal Navier–Stokes/full‐potential method

An improved hybrid method for computing unsteady compressible viscous flows is presented. This method divides the computational domain into two zones. In the inner zone, the Navier–Stokes equations are solved using a diagonal form of an alternating‐direction implicit (ADI) approximate factorisation procedure. In the outer zone, the unsteady full‐potential equation (FPE) is solved. The two zones are tightly coupled so that steady and unsteady flows may be efficiently solved. Characteristic‐based viscous/inviscid interface boundary conditions are employed to avoid spurious reflections at that interface. The resulting CPU times are about 60% of the full Navier–Stokes CPU times for unsteady flows in non‐vector processing machines. Applications of the method are presented for a F‐5 wing in steady and unsteady transonic flows. Steady surface pressures are in very good agreement with experimental data and are essentially identical to the full Navier–Stokes predictions. Density contours show that shocks cross the viscous/inviscid interface smoothly, so that the accuracy of full Navier–Stokes equations can be retained with significant savings in computational time. Copyright © 1999 John Wiley & Sons, Ltd.     

Entropy and vorticity corrections for transonic flows

Different models for inviscid transonic flows are examined. The common assumptions that the flow is isentropic and irrotational are critically evaluated. Entropy and vorticity correction procedures for potential and stream function formulations are presented, together with the details of the treatment of shocks and wakes, and drag and lift calculations. The non-uniqueness problem of the potential formulation is studied using different artificial viscosity forms. Numerical results are compared with Euler solutions.     

跨音速湿空气非平衡凝结流动的数值研究

跨音速流动条件下湿空气中的水蒸气由于快速膨胀而发生非平衡凝结,凝结潜热对跨音速气流进行加热,会显著改变气流的流动特性。通过对商用计算流体动力学软件FLUENT进行二次开发,建立了湿空气非平衡凝结流动的数值求解方法。该方法可用于二维或三维、粘性或无粘、内流或外流的求解中。采用该方法分剐对缩放喷管、透平叶栅以及绕CA-0.1圆弧翼型的湿空气非平衡凝结流动进行了数值分析。计算结果表明：湿空气凝结手l起缩放喷管中的凝结激波、导致叶橱流动中总压降低;对于翼型周围的流动,在相对湿度分别为50％、57．1％、64.1％时,依次计算得到了单激波、五激波、双激波。     

Rotational inviscid flow with circulation zones in a channel of variable cross section

The development of the theory of rotational motion of inviscid fluids for the purposes of describing channel flow encounters certain difficulties in connection with the appearance of viscosity effects near the walls. In the potential-rotational model [1], in which the vorticity is nonzero only in a closed circulation zone surrounded by potential flow, it is assumed that the separation and attachment points are known in advance. For example, for flow around a cavity these points coincide with the extreme corner points of the contour. The problem of determining the vorticity in a closed zone for the potential-rotational model has been investigated in a number of studies [2, 3], etc. In the case of an incompressible fluid the vorticity in the circulation zone is constant for two-dimensional flow and proportional to the distance from the axis for axisymmetric flow. The value of the constant is found from the steady-state condition for the adjoining viscous layers. If the channel walls have a smooth profile without corner points, then for determining the boundaries of the circulation zones additional conditions must be used. This study employs another scheme, in which the vorticity is formed outside the region of flow and in a particular problem is specified in the form of a boundary condition. An analytic solution describing the rotational flow of an inviscid fluid in a channel with a slightly varying cross section is obtained. Three types of entrance flow nonuniformity are considered: 1) uniform shear flow, 2) wake-type flow, and 3) potential flow with a narrow wall boundary layer. Streamline patterns with circulation zones are constructed for flows in diffuser channels with the above-mentioned types of entrance nonuniformity. A model of flow separation in a channel with a turbulent boundary layer on the walls is discussed.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 31–37, March–April, 1985.In conclusion the author wishes to thank E. Yu. Shal'man, A. N. Kraiko, and A. B. Vatazhin for useful discussions and advice.     

Euler analysis of transonic stator-rotor interaction using a finite volume method

A generalized finite volume method that can solve the Euler equations for the stator and rotor parts of stage flow in similar formulations is presented. The method consists of a new moving grid finite volume formulation applied to the rotor region and the existing fixed grid finite volume method used in the stator region, with the data transfer made by an interpolation procedure at the sliding surface in between. The accuracy of the method has been demonstrated on a simple cascade flow before the time-dependent compressor stage flow is fully investigated. The transonic stator-rotor flow interaction is elucidated within the inviscid and rotational flow limit.     

跨音速有旋流动正问题的赝势函数变分有限元法

本文运用赝势函数变分有限元方法数值模拟了绕翼型的跨音速有旋流动。在含有激波的跨音速有旋流动中，势函数已不存在，但为了保留势函数模型在求解方面的优越性，上海大学的刘高联引入了一个通用函数一赝势函数，可以看出该赝势函数保持了势函数的所有好处，又突破了流动有势的限制，是势函数对有旋流动的一个自然的、物理上相容、数学上求解简便的推广，进一步地，刘高联还得到了赝势函数的变分原理族，为变分有限元法求解有旋流动打下了基础。另一方面，为了提高数值求解的收敛性和有效地捕捉流场中的激波，本文还采用了“人工密度”办法。绕翼型的跨音速有旋流动的计算实践证明了赝势函数的有效性。     

The Fascination of Vortex Rings

We present inviscid and viscous models for the formation and propagation of single, and co-axial pairs of, vortex rings. Inviscid flows are based on both thin rings, and thick rings treated by a contour dynamics approach, whilst viscous flows are determined from numerical solutions of the Navier–Stokes equations. A kaleidoscope of different flow behaviours for these axisymmetric flows is presented.     

ANALYSIS/DESIGN CALCULATIONS OF TRANSONIC FLOWS

SUMMARY

Analysis/design calculations of transonic flow are discussed and several improvements are made. The nonisentropic potential method is used to calculate the inviscid transonic flow analysis problem instead of the traditional potential method. An inverse integral 3D boundary layer method is used to calculate the boundary layer in the viscous transonic flow analysis problem. The viscous/inviscid interaction calculations are carried out by a semi-inverse coupling scheme. In design problem calculations, an improved residual-correction method is used. Three individual methods are combined in a global algorithm and computing code. The improvements speed up the convergence, increase applicability and computational efficiency. Some numerical results are given to illustrate that the present method provides an effective engineering tool of high accuracy and efficiency in three dimensional transonic analysis and design situations.     

Asymptotic analysis of transonic flows

In this paper we derive the equations of the second and third approximations for the stream function of two-dimensional and axisymmetric potential transonic flow of an inviscid gas and find their particular solutions corresponding to certain transonic flows.A similar study concerning the second approximation of subsonic and supersonic flow was made by Van Dyke [1] and Hayes [2]. The second approximation for the velocity potential of transonic flow has been examined in detail by Hayes [3]. Euvrard [4, 5] has investigated the asymptotic behavior of transonic flow far from a body, while Fal'kovich, Chernov, and Gorskii [6] have studied the flow in a nozzle throat.The transonic asymptotic analysis for the stream function is presented in this paper.     

Finite volume method for three-dimensional transonic potential flow through turbomachinery blade rows

A finite-volume method has been developed for the calculation of transonic, potential flows through 3-D turbomachinery blades with complex geometries. The exact transonic potential flow equation is solved on a mesh constructed from small volume elements. A transformation is introduced through which cuboids of the physical plane are mapped into computational cubes. Two sets of overlapping volumes are used. While the thermodynamic properties are calculated at the primary volume centres, the flux balance is established on the secondary volumes. For transonic flows an artificial compressibility term (upwind density gradient) is added to density to produce the necessary directional bias in the hyperbolic region. The successive point over-relaxation Gauss-Seidel method has been used to solve the non-linear partial differential equations. Comparisons with experiments and/or other numerical solutions for various turbomachinery configurations show that the 3-D finite-volume approach is a relatively accurate, reliable and fast method for inviscid, transonic flow predictions through turbomachinery blade rows     

A GENERAL FORMULA FOR CALCULATING FORCES ON A 2-D ARBITRARY BODY IN INCOMPRESSIBLE FLOW

In the present paper, a general integral equation is presented to calculate the forces exerted on a two-dimensional (2-D) body of arbitrary shape immersed in unsteady, incompressible flows. By finding the general solutions of a set of Laplace equations with particular boundary conditions, the equation can be simplified to produce a simplified formula for calculating the forces. The simplified formula consists of three parts, representing contributions from different physical phenomena: added mass force and/or inertial force in inviscid flow, the force caused by the deformation of fluid and viscosity and the force caused by the convection of fluid with nonzero circulation. It can be applied to any 2-D arbitrary body in viscous or inviscid, steady or unsteady incompressible flow. As the formula excludes either temporal derivatives of velocity or spatial derivatives of vorticity in the flow field, the numerical errors contained in the numerical solution of velocity and vorticity fields will not be magnified, and therefore the resulting force calculated is more accurate. Most importantly, the formula presents an alternative method for obtaining the added mass of a 2-D body of arbitrary shape accelerating in a fluid. For bodies of simple shape, such as a circle, ellipse and plate, the added masses predicted using the present method are in agreement with that obtained by conventional methods. For bodies of complex shape, the present method only requires the calculation of the first two coefficients of the conformal transformation and cross-sectional area.     

Flow of a viscous heat-conducting gas and binary mixtures in a sink

The class of exact solutions of the one-dimensional Navier-Stokes equations corresponding to gas flows from a spherical source or sink has been investigated analytically and numerically on a number of occasions (see, for example, [1, 2]). Here, the solution for a sink is considered in the presence of heat transfer from the ambient medium. Apart from seeking the solution itself, the object of the investigation was to establish the conditions of transi tion from viscous to inviscid flow in the sink as the Reynolds number tends to infinity. As shown in [3], for zero heat flux at an infinitely remote point there is no such transition for flow in a sink. The sink flow characteristics of a binary gas mixture are investigated in detail. In the transonic flow region an asymptotic solution is obtained.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 56–62, January–February, 1989.     

Vorticity in plane parallel, axially symmetrical or spherical flows of viscous fluid

For viscous (barotropic or incompressible) fluids it is shown that, if the vorticity and the viscous force are orthogonal, vortex lines are convected by a vector field which fits with the velocity field when viscosity vanishes (extension of Helmholtz theorem); it is also found that energy remains constant along the field lines of this vector field (extension of Bernoulli theorem).If, moreover, vorticity and velocity are orthogonal too, the magnitude of the vorticity then behaves as the density of a fluid which flows along streamsheets according to this very same vector field. These properties are mainly encountered for plane parallel flows, axially symmetrical flows, spherical flows, but also for some other miscellaneous flow geometries such as unidirectional or radial flows. The set of the former three flows can even be characterized by these properties; that enhances this set of important flow geometries, avails a general view on vorticity behavior, and explains the great simplicity of vorticity equations in these cases. Numerous examples and comments are given for illustrating.     

Relation between the quasi-cylindrical approximation and the critical classification for swirling flow

It is shown that under the assumption of the quasi-cylindrical approximation with viscous effect the critical flow corresponds to a singularity, as flows approach the critical state from both sides, the radial components of velocity go separately to positive and negative infinity, and that for inviscid flow solution of the quasi-cylindrical approximation can only be trivial solution, i.e. strictly cylindrical flow. It is also shown that in subcritical range the iterative procedure accounting for non-linear effect of equations diverges necessarily. Projects Supported by the Science Fund of the Chinese Academy of Sciences.     

2-D/3-D finite-element solution of the steady Euler equations for transonic lifting flow by stream vector correction

In this paper we study the validation of the new formulation (potential-stream vector) of the steady Euler equations in 2-D/3-D transonic lifting regime flow. This approach, which is based on the Helmholtz decomposition of a velocity vector field, is designed to extend the potential approximation of Euler equations for severe situations such as high transonic or rotational subsonic flows. Different results computed by a fixed point algorithm on the stream vector correction are shown and discussed by comparing them with those obtained by the full potential approach.     

Simulation of blast wave propagation over a cylinder

A numerical study of the interaction of plane blast waves with a cylinder is presented. Computations are carried out for various blast-wave durations and comparisons are obtained with the corresponding results of planar shock-wave. Both inviscid and viscous results based on the solution of the Euler and Navier-Stokes equations are presented. The equations are solved by an adaptive-grid method and a second-order Godunov scheme. The shock wave diffraction over the cylinder is investigated by means of various contour plots, as well as, pressure and skin-friction histories. The study reveals that the blast-wave duration significantly influences the unsteady flow over the cylinder. The differences between the viscous and inviscid results are also discussed. Received 2 March 1996 / Accepted 28 February 1997     

On the accuracy and efficiency of CFD methods in real gas hypersonics

A study of viscous and inviscid hypersonic flows using generalized upwind methods is presented. A new family of hybrid flux-splitting methods is examined for hypersonic flows. The hybrid method is constructed by the superposition of the flux-vector-splitting (FVS) method and second-order artificial dissipation in the regions of strong shock waves. The conservative variables on the cell faces are calculated by an upwind extrapolation scheme to third-order accuracy. A second-order-accurate scheme is used for the discretization of the viscous terms. The solution of the system of equations is achieved by an implicit unfactored method. In order to reduce the computational time, a local adaptive mesh solution (LAMS) method is proposed. The LAMS method combines the mesh-sequencing technique and local solution of the equations. The local solution of either the Euler or the NAVIER-STOKES equations is applied for the region of the flow field where numerical disturbances die out slowly. Validation of the Euler and NAVIER-STOKES codes is obtained for hypersonic flows around blunt bodies. Real gas effects are introduced via a generalized equation of state.