Nonlinear capillary free-surface flows

Capillary free-surface flows are considered. The fluid is taken to be inviscid and incompressible and the flow to be irrotational. Particular attention is devoted to two-dimensional flows for which the free surfaces intersect rigid walls. These include cavitating flows and local flows at the front of a small object (probe or insect) moving at the surface of a fluid. A general study of the effect of surface tension on the possible singularities which can occur at the separation points is presented. The results confirm and generalise previous findings on the subject.     

Concerning inviscid solutions for large-scale separated flows

Summary The large-scale separated eddies set up behind a bluff body at high Reynolds number are considered, for steady laminar planar flow. The main eddies are massive and are controlled predominantly by inviscid mechanics, with uniform vorticity inside. Analytical and computational solutions of the massive-eddy (vortex-sheet) problem are then described. A further possibility studied is that, even with lateral symmetry assumed, there may still be an extra degree of nonsymmetry of skewing with respect to the streamwise direction. Small-scale separations, where a Benjamin-Ono equation also possibly yielding nonsymmetric solutions can come into play, are discussed briefly.     

Fully non-linear two-layer flow over arbitrary topography

Steady, two-dimensional, two-layer flow over an arbitrary topography is considered. The fluid in each layer is assumed to be inviscid and incompressible and flows irrotationally. The interfacial surface is found using a boundary integral formulation, and the resulting integrodifferential equations are solved iteratively using Newton's method. A linear theory is presented for a given topography and the non-linear theory is compared against this to show how the non-linearity affects the problem.     

Note on a paper by Sinha and Odgaard ``Application of conformal mapping to diverging open channel flow''''

A steady gravity-free two-dimensional potential flow separating a parent channel and an offtake channel is studied by conformal mapping of a half-strip in the Zhukovskii domain onto a strip with a cut in the complex potential domain. A parameter of the mapping and the equation of a free streamline are found by computer-algebra routines.     

Waves and singularities in nonlinear capillary free-surface flows

The problem is a free-surface flow of a fluid, emerging from a semi-infinite container. The fluid is assumed to be inviscid, incompressible and the flow to be two dimensional and irrotational. When surface tension is neglected the free surface leaves the wall of the container tangentially. We show that when surface tension is taken into account, there is, in general, a train of waves on the free surface and a discontinuity in slope where the free surface separates from the wall of the container. These new solutions include, as particular cases, previously obtained solutions for which the free surface is waveless in the far field. Although the calculations are presented for a special flow configuration, the results are general and apply to other potential free surface flows where a free-surface intersects a rigid wall.     

Analysis of a bifurcation problem in marginally separated laminar wall jets and boundary layers

Summary.  The paper deals with the steady laminar viscous inviscid interaction which arises if a wall jet is subjected to a rapid deflection caused for example by the presence of a ramp. The deflection angle is taken such that the flow is at the verge of separation or even slightly, i.e. marginally, separated. Although the flow entering and leaving the interaction region is assumed to be strictly two-dimensional fully three-dimensional disturbances are accounted for in the treatment of the local interaction process. Similar to the related boundary layer problem studied by Braun and Kluwick [3] these disturbances are found to be governed by a nonlinear integro-differential equation. In contrast to this case, however, the induced pressure disturbances now depend on the local curvature of the streamlines only rather than on global flow properties. Special emphasis is placed on the asymptotic analysis of a bifurcation problem associated with the non-uniqueness of the solutions of the interaction equation. Received September 2, 2002 Published online: April 17, 2003     

Two-layer critical flow over a semi-circular obstruction

Steady, trans-critical flow of a two-fluid system over a semi-circular cylinder on the bottom of a channel is considered. Each fluid is assumed to be inviscid and incompressible and to flow irrotationally, but the fluids have different densities, so that one flows on top of the other. Consequently, a sharp interface exists between the fluids, in addition to a free surface at the top of the upper fluid. Trans-critical flow is investigated, in which waves are absent from the system, but the upstream and downstream fluid depths differ in each fluid layer. The problem is formulated using conformal mapping and a system of three integrodifferential equations, and solved numerically with the aid of Newton's method. The free-surface shape and that of the interface are obtained along with the Froude numbers in each fluid layer. Results of computation are presented and discussed.     

Analytical series solutions for three-dimensional supercritical flow over topography

An analytical series solution method for three-dimensional, supercritical flow over topography is presented. Steady, nonlinear solutions are calculated for a single layer of inviscid, constant-density fluid that flows irrotationally over an obstacle that varies significantly in the x-, y- and z-directions. Accurate series solutions for the free surface and a series of stream tubes throughout the flow region are calculated to demonstrate the three-dimensional properties of the problem. These solutions provide valuable insight into the three-dimensional interactions between the fluid and obstacle which is impossible to gain from any two-dimensional model. The model is described by a Laplacian free-boundary problem with fully nonlinear boundary conditions. The solution method consists of iteratively updating the location of the free surface (on top of the fluid) using a cost function which is derived from the Bernoulli equation. Root-mean-square errors in the boundary conditions are used as convergence criteria and a measure of the accuracy of the solution. This method has been used to solve the two-dimensional version of this problem in the past. Here, we detail the extensions required for three-dimensional flow.     

Waves generated by a source below a free surface in water of finite depth

The free-surface flow due to a submerged source in water of finite depth is considered. The fluid is assumed to be inviscid and incompressible. The problem is solved numerically by using a boundary integral equation formulation due to Hocking and Forbes [6]. The numerical results show that there is a train of waves on the free surface in accordance with the results of Mekias and Vanden-Broeck [5]. For small values of the Froude number, the amplitude of the waves is so small that the free surface is essentially flat in the far field. These waveless profiles agree with the calculations of Hocking and Forbes [6].     

Exponential similarity free convection boundary layers over curved surfaces

Summary This paper reports novel similarity solutions of exponential type for the steady free convection boundary-layer flow over two-dimensional heated bodies of arbitrary surfaces. The existence of a two-parameter family of curved surfaces is shown to exist. The geometrical characteristics of these surfaces, described in terms of elementary transcendental functions, are discussed in detail. Compared to the well known body shapes which permit similar free convection flows of power-law type, substantial differences (as cusps at the leading edge, concave shapes going over in horizontal plateaus, etc.) have been found.     

Determining wake flow parameters for two counterphase synchronized von Kármán vortex streets from configuration stability analysis

The problem of determining the parameters of near wake flow past a pair of cylindrical bluff bodies from one-point spectra of the velocity pulsation is solved in the limit of an inviscid incompressible flow. For this purpose, the stability of wake configuration with respect to infinitesimal perturbations of the equilibrium localization of the vortices is analyzed within the framework of a flow model of two counterphase synchronized von Kármán vortex streets. A necessary condition for the flow stability is determined.     

Upstream influence in hypersonic aerodynamics over sharp thin and thick bodies

This theoretical analysis is for a sharp aligned body with a leading-edge shock wave, within a steady hypersonic free stream. The flow field in the shock layer containing a viscous and an inviscid layer is taken to be steady, laminar and two-dimensional, for a perfect gas without real and high-temperature gas effects. The hypersonic shock layer under different conditions is analyzed in order to offer an analytical explanation of the behavior of upstream influence.     

Deep- and shallow-water slamming at small and zero deadrise angles

This paper reviews and extends theories for two classes of slamming flows resulting from the violent impact of bodies on half-spaces of inviscid fluid. The two configurations described are the impact of smooth convex bodies, and of non-smooth but flat-bottomed bodies, respectively. In each case, theories are presented first for small penetration depths in finite- or infinite-depth fluids (which we call Wagner flows), and secondly when the penetration is comparable to the fluid depth (which we call Korobkin flows). We also discuss the transition from Wagner flow to Korobkin flow.     

Use of bluff body for enhancing submicron particle agglomeration in plasma field

A bluff body was installed to generate a vortex shedding in gas flow under nonthermal plasma field. Various shapes of bluff body were analyzed using computational fluid dynamics for their ability to enhance submicron particle agglomeration. The cylindrical bluff body produced the lowest pressure drop and the plate bluff body showed the widest amplitude of vortex shedding. In the experiment, exhaust gas with a velocity of 1–3?m/s were fed into the test section. The electrical pulse peak voltage was 35?kV, 10?kHz. The bluff body improved the reduction efficiency by 27% and 17% for flat plate and cylindrical bluff bodies, respectively, relative to no-bluff body.     

Uniqueness of the bounded flow solution in aerodynamics

We discuss uniqueness for steady incompressible inviscid flows past a body with a sharp trailing edge TE, with particular regard to multiconnected (toroidal) 3D wing configurations. Boundedness of the velocity field at TE is enforced by means of a singularity removal principal (Kutta condition). The resulting bounded flow solution is unique for 2D airfoils and 3D conventional wings. For toroidal bodies the flow depends on the available eigensolution which, however, has no direct influence on the lift. In this multiconnected case uniqueness of the bounded solution is shown to depend on the topology of the trailing edge.     

Retrieval of the shape of the bottom surface of a channel when the free surface profile is given

Two-dimensional, steady, inviscid, irrotational, incompressible and gravity influenced free surface fluid flow in a channel, over a step of an unknown shape, was investigated when the free surface profile is given. A combination of the boundary integral method, the minimal height method and the application of a minimum plot procedure were employed in order to give an optimal solution to the problem.     

Circle theorems for inviscid steady flows

General circle theorems which localize the complex eigenfrequencies arising in the linear stability analysis of conservative steady flows are given. Howard's circle theorem for incompressible plane parallel flow is contained as a special case. Two applications are considered: swirling flow of an inviscid incompressible fluid, and rotating flow of an inviscid, incompressible, perfectly conducting magnetofluid with an axial magnetic field. Circle theorems are obtained for the complex eigenfrequencies of any normal mode.     

The wave resistance of a two-dimensional body moving forward in a two-layer fluid

A two-dimensional body moves forward with constant velocity in an inviscid, incompressible fluid under gravity. The fluid consists of two layers having different densities, and the body is totally submerged in one of them. The resulting fluid motion is assumed to be steady state in a coordinate system attached to the body. The boundary-value problem for the velocity potential is considered in the framework of linearized water-wave theory. The asymptotics of the solution at infinity is obtained with the help of an integral representation, based on the explicitly known Green function. The theorem of unique solvability is formulated, and the method applied to prove it is briefly explained (the detailed proof is given in another work). An explicit formula for the wave resistance is derived and discussed. A numerical example for the wave resistance serves to illustrate the so-called dead-water phenomenon.     

A boundary integral method for gravitational fluid flow over a semi-circular obstruction using an interpolative technique

A boundary integral technique is developed to study the behaviour of a steady, two-dimensional free surface flow of an incompressible, irrotational and inviscid fluid over a submerged semi-circular obstruction in the presence of gravity. The solution technique is different to that employed by many of the previous research workers since it involves the application of the Riemann–Hilbert problem in the derivation of the nonlinear boundary integral–differential equations. The boundary integral equations are solved using piecewise constant and linear interpolative techniques for the fluid velocity on both the solid boundary and the free surface for various values of the upstream Froude number and the radius of the semi-circular obstruction. An investigation into the numerical accuracy of the interpolation techniques is employed. It is found that it is difficult to obtain a solution when the non-dimensional radius of the semi-circular obstruction is large and hence a hybrid technique is developed which is capable of computing the free surface profiles for all values of the radius of the semi-circular obstruction. Also by considering the local Froude number we have found that the fluid flow can become subcritical, i.e. the local Froude number is less than unity, in the vicinity just above the obstacle but no waves are found to be present on the free surface.