The Onset of Fluid Rotation in a Thermogravitational Boundary Layer with Local Cooling of the Free Surface

Axisymmetric regimes of flows of an inhomogeneous fluid in the boundary layer near a free surface are calculated for a nonuniform temperature distribution on this surface. For the fluid motion equations written in the Oberbeck-Boussinesq approximation, the leading terms of asymptotic expansions of solutions of a steady-state problem are constructed. It is shown that in the presence of local cooling of the free surface and a rising outer fluid stream, as a result of a bifurcation, a pair of rotational regimes may develop in a thin boundary layer near the free surface, with no rotation observed outside this layer. No bifurcation of rotation was detected in the case of local heating of the free surface.     

Laminar swirling power law non-newtonian fluid jet impinging on a normal plane

An analysis is presented for a steady, laminar, incompressible, swirling, Ostwald—De Waele type non-Newtonian fluid jet impinging normally over a horizontal plane with a free surface. A similarity solution is obtained in a region of radial distance away from the central stagnation point. Numerical solutions for the radial and swirling velocities have been obtained for the flow behavior index, n, varying from 0.1 to 2.5. Expressions for the free surface radial velocity, growth of the free surface and skin friction coefficient are given.     

Application of a vortex method to free surface flows

Vortex methods have found wide applications in various practical problems. The use of vortex methods in free surface flow problems, however, is still very limited. This paper demonstrates a vortex method for practical computation of non-linear free surface flows produced by moving bodies. The method is a potential flow formulation which uses the exact non-linear free surface boundary condition at the exact location of the instantaneous free surface. The position of the free surface, on which vortices are distributed, is updated using a Lagrangian scheme following the fluid particles on the free surface. The vortex densities are updated by the non-linear dynamic boundary condition, derived from the Euler equations, with an iterative Lagrangian numerical scheme. The formulation is tested numerically for a submerged circular cylinder in unsteady translation. The iteration is shown to converge for all cases. The results of the unsteady simulations agree well with classical linearized solutions. The stability of the method is also discussed.     

Mixing layer under a free surface

In the long-wave approximation, the flow of a homogeneous fluid with a free surface in the gravity field is considered. Mathematical models of the surface turbulent layer in shear flows are derived. Steady solutions of the problem of evolution of the mixing layer under the free surface and formation of a surface turbulent jet are constructed. In particular, the problem of the structure of a turbulent bore in a supercritical flow is solved, and the conditions for the formation of a local subcritical zone ahead of the obstacle are studied.     

Non-Rayleigh Convection Caused by Ground Surace Topography

Non-Rayleigh convection is studied in aquifers, where the lateral variations of the temperature field are caused by the ground surface. Analytical solutions for the stream function, the Darcy velocities and the excess water pressure are found for a confined, rectangular and homogeneous aquifer situated at a given depth. The Darcy velocity field and the excess pressure are also found for an unconfined aquifer with a prescribed fluid potential at the top surface of the aquifer. Approximate solutions for excess pressure and Darcy velocities are found for the unconfined aquifer with a free surface. General estimates for the convective Darcy velocities are derived from these solutions. It is also shown that non-Rayleigh flow is unlikely in an unconfined aquifer.     

Small‐Time Expansion of Wave Motion Generated by a Submerged Sphere

A nonlinear problem of motion of a solid sphere near a free surface of an infinitely deep fluid is considered. For the case of motion with a constant acceleration starting from rest, the solution is studied using a smalltime expansion. Expansion coefficients up to the fourth power inclusive are found for the free surface elevation and for the force acting on the sphere. The solutions for linear and nonlinear conditions on the free surface are compared.     

Change-of-type behavior in viscoelastic slender jet models

In this paper a slender jet model of viscoelastic fluids which is asympotically derived from the full free surface boundary-value problem. The model system consists of four coupled quasi-linear differential equations in one space dimensions, where the nonlinear characteristics are given in closed form. Two characteristics are always real, two others may be real or complex, leaving open the possibility for change-of-type from hyperbolic to mixed elliptic/hyperbolic type. We proceed to exhibit exact solutions (constant, steady time dependent and space-time dependent) along which this model system undergoes a variety of change-of-type phenomena. Viewed purely as a one-dimensional (1-D) model for change-of-type, we explain the significance of type for the stability of these solutions and describe the numerical implications for each type. We also explain the physical significance of these model phenomena with respect to the original 3-D system, since these asymptotic equations are no longer valid once small-scale instabilities develop. Remarkably, these special solutions of the model system that exhibit change-of-type correspond to exact solutions of the 3-D Maxwell model with cylindrical free surface. The 1-D model equations, however, are not an invariant reduced system of the full 3-D free surface Maxwell model, so that the change-of-type exhibited here in the 1-D model is not directly responsible for a 3-D free surface change-of-type. Regularizations of this model as a catastrophic change-of-type develops are suggested.Research support is gratefully acknowledged from the Air Force Office of Scientific Research, Grant No. 88-0164.     

The characteristic‐based split (CBS) meshfree method for free surface flow problems in ALE formulation

A semi‐implicit characteristic‐based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.     

On the existence of surface wave solutions in piezoelectric crystals an example of non-existence

According to general theory, a semi-infinite piezoelectric crystal with mechanically free surface and an adjoining medium with zero dielectric constant, may or may not allow a surface wave solution. Examples where surface wave solutions are not allowed, are not known, and one might think that solutions actually always exist. In the present paper this problem is put to rest by an explicit example of non-existence.     

A marker-and-cell approach to viscoelastic free surface flows using the PTT model

This work is concerned with the development of a numerical method capable of simulating two-dimensional viscoelastic free surface flows governed by the non-linear constitutive equation PTT (Phan-Thien–Tanner). In particular, we are interested in flows possessing moving free surfaces. The fluid is modelled by a marker-and-cell type method and employs an accurate representation of the fluid surface. Boundary conditions are described in detail and the full free surface stress conditions are considered. The PTT equation is solved by a high order method which requires the calculation of the extra-stress tensor on the mesh contour. The equations describing the numerical technique are solved by the finite difference method on a staggered grid. In order to validate the numerical method fully developed flow in a two-dimensional channel was simulated and the numerical solutions were compared with known analytic solutions. Convergence results were obtained throughout by using mesh refinement. To demonstrate that complex free surface flows using the PTT model can be computed, extrudate swell and a jet flowing onto a rigid plate were simulated.     

Free surface asymptotics for thermocapillary flow at high marangoni numbers

Formal asymptotic expansions of the solution of the steady-state problem of incompressible flow in an unbounded region under the influence of a given temperature gradient along the free boundary are constructed for high Marangoni numbers. In the boundary layer near the free surface the flow satisfies a system of nonlinear equations for which in the neighborhood of the critical point self-similar solutions are found. Outside the boundary layer the slow flow approximately satisfies the equations of an inviscid fluid. A free surface equation, which when the temperature gradient vanishes determines the equilibrium free surface of the capillary fluid, is obtained. The surface of a gas bubble contiguous with a rigid wall and the shape of the capillary meniscus in the presence of nonuniform heating of the free boundary are calculated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 61–67, May–June, 1989.     

URANS simulations for a high-speed transom stern ship with breaking waves

An exploratory study of high-speed surface ship flows is performed to identify modelling and numerical issues, to test the predictive capability of an unsteady RANS method for such flows, to explain flow features observed experimentally, and to document results obtained in conjunction with the 2005 ONR Wave Breaking Workshop. Simulations are performed for a high-speed transom stern ship (R/V Athena I) at three speeds Froude number (Fr) = 0.25, 0.43 and 0.62 with the URANS code CFDSHIP-IOWA, which utilizes a single-phase level set method for free surface modelling. The two largest Fr are considered to be high-speed cases and exhibit strong breaking plunging bow waves. Structured overset grids are used for local refinement of the unsteady transom flow at medium speed and for small scale breaking bow and transom waves at high-speeds. All simulations are performed in a time accurate manner and an examination of time histories of resistance and free surface contours is used to assess the degree to which the solutions reach a steady state. The medium speed simulation shows a classical steady Kelvin wave pattern without breaking and a wetted naturally unsteady transom flow with shedding of vortices from the transom corner. At higher speeds, the solutions reach an essentially steady state and display intense bow wave breaking with repeated reconnection of the plunging breaker with the free surface, resulting in multiple free surface scars. The high-speed simulations also show a dry transom and an inboard breaking wave, followed by outboard breaking waves downstream. In comparison to an earlier dataset, resistance is well predicted over the three speeds. The free surface predictions are compared with recent measurements at the two lowest speeds and show good agreement for both non-breaking and breaking waves.     

Exact solutions of thin film flows

This article studies the analytical solutions for two thin film flow problems on a moving belt. The reduction of the equations follows from their Lie point symmetry generators and conservation laws which are valid for the considered boundary conditions also. The solutions for the two problems are developed using the correct and nonlinear boundary condition for the free surface. Mathematica is adopted for some of the analysis.     

An explicit formulation for the evolution of nonlinear surface waves interacting with a submerged body

An explicit formulation to study nonlinear waves interacting with a submerged body in an ideal fluid of infinite depth is presented. The formulation allows one to decompose the nonlinear wave–body interaction problem into body and free‐surface problems. After the decomposition, the body problem satisfies a modified body boundary condition in an unbounded fluid domain, while the free‐surface problem satisfies modified nonlinear free‐surface boundary conditions. It is then shown that the nonlinear free‐surface problem can be further reduced to a closed system of two nonlinear evolution equations expanded in infinite series for the free‐surface elevation and the velocity potential at the free surface. For numerical experiments, the body problem is solved using a distribution of singularities along the body surface and the system of evolution equations, truncated at third order in wave steepness, is then solved using a pseudo‐spectral method based on the fast Fourier transform. A circular cylinder translating steadily near the free surface is considered and it is found that our numerical solutions show excellent agreement with the fully nonlinear solution using a boundary integral method. We further validate our solutions for a submerged circular cylinder oscillating vertically or fixed under incoming nonlinear waves with other analytical and numerical results. Copyright © 2007 John Wiley & Sons, Ltd.     

Solutions of Green's function for Lamb's problem of a two-phase saturated medium

The solutions of Green's function are significant for simplification of problem on a two-phase saturated medium. Using transformation of axisymmetric cylindrical coordinate and Sommerfeld's integral, superposition of the influence field on a free surface, authors obtained the solutions of a two-phase saturated medium subjected to a concentrated force on the semi-space.     

Asymptotics of axially symmetric fluid flows with a free boundary in the case of dwindling viscosity

For high Reynolds numbers asymptotic expansions are constructed of the solution of the axially symmetric wave problem on the surface of a viscous incompressible fluid of infinite depth under the assumption that the tangential stresses on the free surface are of the order 0(1/Re). The principal terms of the asymptotic expansion are solutions of linear partial differential equations. The obtained result is then adapted to the case in which the fluid fills a bounded region whose boundary is a free surface. Some examples are given.     

Solitary waves in a viscous liquid film flowing down a thin vertical cylinder

Several equations to describe the flow of a viscous liquid film on a thin cylinder are derived. The solitary-wave solutions to these equations are studied. The families of solutions are constructed for the first two eigenvalues that correspond to single-humped and double-humped waves. It is found that these families become similar as the similarity parameter increases. The dependencies of phase velocities and wave amplitudes on the free parameters of the problem are analyzed. The resulting solutions are compared with solitary waves in films on a flat surface.     

部分滑移边界条件下自由液面流动问题研究

数值方法进行相关问题的研究就以圆柱形储液罐为例, 考虑部分滑移边界条件, 对所得的高 度非线性微分方程进行了数值求解, 讨论了贮液腔体内液面接触线和液体高度线, 及部分滑 移条件对流体层微观半径的影响; 所得结论对自由液面晃动问题数值研究中的边界条件处理 有参考意义.     

Numerical simulation of fully non-linear steady free surface flow

The fully non-linear free surface potential flow past a 2D non-lifting body is computed. The numerical method is based on the simple layer integral formulation; the non-linear solution is obtained by means of an iterative procedure. Under some hypotheses, viscosity effects at the free surface are considered. All the numerical results obtained have been tested against analytical solutions and experimental results.     

Elastic reciprocity and symmetry constraints on the stress field due to a surface-parallel distribution of dislocations

Elastic reciprocity and geometric symmetry are used to constrain the expressions for stresses due to introduction of line dislocations near a half-space surface. Specifically, a relationship is shown to exist between the changes induced by dislocations of orthogonal Burgers vectors (normal and parallel to the free surface). These results are used to address inconsistencies of solutions in the literature.