Flow past a plate with a moving surface

On the basis of a numerical solution of the unsteady Navier-Stokes equations, the flow past a finite plate with an upstream-moving surface is investigated. For the Reynolds numbers Re =10²−10⁴, the flow past the plate is analyzed as a function of the relative plate surface velocity. On the basis of this analysis a limiting mathematical model of the flow as Re → ∞ is proposed.     

Flow past a plate with an upstream-moving surface

The numerical solution of the problem of flow past a plate whose surface travels in the opposite direction to the stream is obtained under the assumption that the surface velocity is higher than the free-stream velocity. The limiting flow diagram as t → ∞ is predicted and justified.     

Flow past a circular cylinder over a free surface: Interaction between the near wake and the free surface deformation

Air-flow around a circular cylinder placed above a free surface and liquid flow under the free surface were investigated experimentally in a wind/wave tunnel. The cylinder spanned the tunnel test-section and was oriented normal to the freestream direction. The main objective of this study was to investigate the interaction of the cylinder wake with the free surface. The flow structure was analyzed for various gap widths, H, between the cylinder and the free surface using a digital particle image velocimetry (PIV) system with a spatial resolution of 2048×2048 pixels. The Reynolds number based on the cylinder diameter was 3.3×10³. For each experimental condition, 400 instantaneous velocity fields were measured and ensemble-averaged to obtain spatial distributions of the mean velocity and turbulence statistics. The results showed that the cylinder near-wake inclined upward due to the influence of the free surface elevation. Vortices were shed, even at a small gap ratio of H/D=0.25, where D is the cylinder diameter. Strong jet-like flow appeared in the gap beneath the cylinder. At a gap ratio of H/D=0.50, the jet flow exhibited a quasi-periodic vibration with a period of 2–3 s. The free surface deformation was caused by the pressure difference in the air-flow immediately above it. As the gap ratio increased, the inclination angle of the wake and the height of the free surface elevation decreased gradually. The liquid flow under the free surface followed a convective flow motion, and the range of the convection depended on the gap width between the cylinder and the free surface.     

Parameters of spatial flow past a hydrofoil near the free surface of a ponderable liquid

An approach is presented to the investigation of spatial flow past an submerged hydrofoil in the presence of a ponderable liquid. The approach uses the method of singularities, which represents the influence of the free surface of the water by the influence of an unbounded layer of sources. The linear solution is completely found for the problem of determining the wave production and downwash behind an submerged hydrofoil with given parameters of the eddy system. An estimate is made of the influence on the flow characteristics of nonlinear factors associated with the change produced by the wave production in the form of the free eddies leaving the hydrofoil. The distribution of the downwash in the flow behind the hydrofoil is found and recommendations given for the arrangement of a hydrofoil system with positive interaction of the elements. The results of the calculation are compared with experiment.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza., No. 2, pp. 54–62, March–April, 1979.     

Flow past a suddenly cooled horizontal plate

The problem of unsteady laminar, incompressible free convection above a horizontal semi-infinite flat plate is studied theoretically. It is assumed that for timet<0 the plate is hotter than its surroundings and at timet=0 the plate is suddenly cooled to the same temperature of its surroundings. Three solutions of the momentum and energy equations are obtained, namely

1. an analytical solution which is valid for small time,
2. an asymptotic analytical solution which is valid for large time, and
3. a numerical solution which matches these two limiting analytical solutions.

It is found that the numerical solution matches the small and large time solutions accurately. Finally, the variation of the velocity, temperature, skin friction and heat transfer on the plate with time are discussed.     

Flow past a plate lattice with developed cavitation

Problems of streamline cavitation flow past a lattice were studied in [1–8] using the Kirchhoff scheme. In this scheme the magnitude of the velocity at the free surface is equal to the stream velocity behind the lattice, and the cavitation number is zero (for a lattice the relative velocity and the cavitation number are defined from the stream velocity behind the lattice). In [4, 7] a solution is given of the problem of flow past a lattice using a scheme with an Efros-Gilbargreturn streamline, which permits considering arbitrary cavitation numbers; however, a unique solution is not given. Some other streamline schemes are mentioned in [8].In the following we consider the cavitational flow of an ideal incompressible inviscid and weightless fluid past an infinite lattice of flat plates, using the streamline wake model previously utilized by Wu [9] in studying cavitational flow past an isolated obstacle. In accordance with this model, the streamlines which separate from the body and bound the cavity behind it pass into two curvilinear infinitely long walls, along which the pressure increases and approaches the pressure in the undisturbed stream.It is further assumed that in the hodograph plane there corresponds to the curvilinear walls a cut along some line and that the complex potential takes the same values at points lying on opposite sides of the cut. In particular, at the points of contact of the streamlines with the curvilinear walls the complex potential is the same. In the Wu scheme the latter condition leads to vanishing of the velocity circulation along the contour CABC₁ (Fig. 1).In conclusion the author wishes to thank N. V. Yurtaeva for the accurately performed numerical work.     

Magnetohydrodynamic unsteady free convection past a hot vertical plate

The free convection flow of an electrically conducting liquid from an infinite plate has been studied in the presence of a uniform magnetic field. General expressions for the velocity field, induced magnetic field, skin-friction and temperature distribution have been obtained when the plate is a perfect conductor and its temperature varies with the law t ⁿ e ^at . The results have been presented through some graphs and tables with the magnetic Prandtl number unity as its value.     

Flow past an impulsively started vertical plate with variable surface temperature and mass flux

Finite-difference solution of the transient natural convection flow of an incompressible viscous fluid past an impulsively started semi-infinite vertical plate with variable surface temperature and mass flux is presented here. The Velocity profiles are compared with exact solution and are found to be in good agreement. The steady-state velocity, temperature and concentration profiles are shown graphically. It is observed that there is a rise in the velocity due to the presence of a mass diffusion. The local as well as average skin-friction, Nusselt number and Sherwood number are shown graphically. Received on 27 May 1998     

Transient conjugate heat transfer from a hemispherical plate during free liquid jet impingement on the convex surface

This paper considers the analysis of transient heating of a hemispherical solid plate of finite thickness during impingement of a free liquid jet. A constant heat flux was imposed at the inner surface of the hemispherical plate at t = 0 and heat transfer was monitored for the entire duration of the transient until a steady state condition was reached. Calculations were done for Reynolds number (Re) ranging from 500 to 1,500 and dimensionless plate thicknesses to nozzle diameter ratio (b/d _n) from 0.083 to 1.5. Results are presented for local and average Nusselt number using water as the coolant and various solid materials such as silicon, constantan, and copper. It was detected that increasing the Reynolds number decreases the time for the plate to achieve the steady-state condition. Also, a higher Reynolds number increases the Nusselt number. Hemispherical plate materials with higher thermal conductivity maintain lower temperature non-uniformity at the solid–fluid interface. Increasing the plate thickness decreases the maximum temperature in the solid and increases the time to reach the steady-state condition.     

Source below the free surface of a heavy liquid

The problem of the motion of a source under the free surface of an infinitely deep heavy liquid has been studied by Keldysh [1] under the assumptions used in the theory of small amplitude waves. However, these assumptions are no longer valid [2] for large Froude numbers F.A method using only one of the four assumptions of small amplitude wave theory (the assumption that the absolute value of the velocity at the free surface is nearly constant) was described in [2], In the following, this method is used to construct a solution of the problem which becomes exact as F . When F is not large our results are close to those of Keldysh if the source intensity is low. For F=0, both methods lead to exact results.     

Propulsive performance of a flapping plate near a free surface

The propulsive performance, i.e., the time-averaged thrust coefficient or the propulsive efficiency, of a flapping flat plate advancing near an otherwise quiescent free surface (liquid–gas interface) with Re of 1000, Fr of 0.2 and 0.8, and various submergence depths is numerically investigated by employing an adaptive Cartesian cut-cell/level-set method. The flapping kinematics parameters excluding the pitch-leading-heave phase angle were fixed as those commonly seen in literature. Results show that for submergence depth larger than the heave amplitude, the propulsive performance peaks at a smaller pitch-leading-heave phase angle with a shallower submergence for Fr of 0.2 but at the same phase angle for Fr of 0.8. Proximity to the free surface enhances the peak propulsive performance for Fr of 0.2 but the influence is minor for Fr of 0.8. The propulsive performance with Fr of 0.2 increases with decreasing chord-normalized submergence depth for the pitch-leading-heave phase angle smaller than 100°. The trend is reversed for the pitch-leading-heave phase angle larger than 100°. However, the propulsive performance with Fr of 0.8 hardly depends on the chord-normalized submergence depth. For submergence depth equal to the heave amplitude, the temporal variation in the thrust coefficient exhibits characteristics inherently different from those with other submergence depths for Fr of 0.2. Also, the time-averaged thrust coefficient exhibits a unique variation with the pitch-leading-heave phase angle. However, the various characteristics of the propulsive performance are similar to those with other submergence depths for Fr of 0.8. For submergence depth smaller than the heave amplitude and Fr of 0.2, the propulsive performance gains much from exposure of the upper surface of the plate to the gas phase. The efficiency enhancement is linked to the weakening of the leading edge vortices. A second harmonic with significant amplitude is found in the upstream wave for Fr of 0.2 with a typical pitch-leading-heave phase angle.     

Flow of a liquid metal coolant past a heat generating cylinder

The coupled problem whereby a solid heat generating cylinder is being cooled in steady state by a coolant in potential flow is investigated. An analytical technique for determining the temperature distributions in the solid and the fluid is presented. Numerical studies for six Péclet numbers (0.9<Pe<11.3) and three thermal conductivity ratios (0.31<K<3.1) were carried out.The surface hot-spot temperature and center temperature are presented graphically as functions of the Péclet number with the thermal conductivity ratio as a parameter. The average Nusselt number is found to be proportional to the Péclet number to approximately the one-half power. For the special case of constant surface temperature (uncoupled problem), the variation of local Nusselt number with angle measured from the forward stagnation point is in excellent agreement with the result presented by Grosh and Cess [6].Nomenclature ce _m(, –q) Mathieu function, periodic - D _n Fourier coefficient for solid temperature distribution - E _n Fourier coefficient for fluid temperature distribution - E() a term defined by equation (12), degree - F() a term defined by equation (13), degree - Fek _m(z, –q) modified Mathieu function, non-periodic - Fek _m(z, –q) derivative of Fek _m(z, –q) - h local heat transfer coefficient, energy/time area degree - average heat transfer coefficient, energy/time area degree - h _m mean heat-transfer coefficient, energy/time area degree - k _f thermal conductivity of fluid, energy/time length degree - K thermal conductivity ratio, k _f/k _s - k _s thermal conductivity of solid, energy/time length degree - Nu local Nusselt number, 2Rh/k _f - average Nusselt number defined by equation (55) - (Nu)_m mean Nusselt number defined by equation (57) - Pe Péclet number, 2RU/ _f - Q rate of heat generation per unit volume, energy/time volume - q parameter of Mathieu function, (Pe/4)² - q normal heat flux, energy/time area - R cylinder radius, length - Re Reynolds number, 2R/ - r radial position variable, length - T temperature, degree - T ₀ constant surface temperature, degree - T temperature of fluid at infinity, degree - T _e temperature at center of cylinder, degree - T _f temperature of fluid, degree - T _s temperature of solid, degree - T _w surface temperature, degree - surface hot-spot temperature, degree - reduced temperature, (T–)/E(1) - U approach velocity of flowing fluid, length/time - v velocity component in the direction, length/time - v _r velocity component in the r direction, length/time - z logarithm of Greek symbols _f thermal diffusivity of the fluid, (length)²/time - reduced radius, r/R - angular position variable measured from the trailing stagnation point, radians - kinematic viscosity, (length)²/time - angular position variable measured from the forward stagnation point, degree     

Effect of buoyancy on the free surface flow past a permeable bed

Laminar steady free surface flow having one permeable bounding wall is investigated in the presence of buoyancy force. The experimental results of Rajasekhara [1] were found to be in good agreement with our theoretical results based on a model which admits slip-velocity at the porous material. The effect of buoyancy force is to increase the velocity distribution in the case of greater heat addition (N_o>0) and to decrease it by a greater cooling (N_o<0). As a result, the mass flow rate increases and the friction factor decreases for N_o>0 and the opposite is true for N_o<0. We further find that the effect of buoyancy force on the temperature distribution is to increase its magnitude. In particular, we find that the rate of heat transfer at its nominal surface is increased in the case of heating (N_o>0) of flow.     

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.     

A vortex below the free surface of a heavy liquid

The problem of the motion of a vortex below the free surface of an infinitely deep heavy liquid was first studied by Sretenskii in 1933 (see [1]). Soon thereafter Keldysh [1] gave a simpler solution of the problem of flow with a free surface past an isolated singularity. The basis of the studies of Sretenskii and Keldysh was the theory of small amplitude waves, which, as will be shown, is no longer valid for large Froude numbers F.The method presented below permits obtaining a solution which becomes exact as F. When F is not large, this solution agrees approximately with the previous solutions if the vortex intensity is low.     

Combined forced and free convection MHD row past a semi-infinite vertical plate

Effects of a transversely applied magnetic field on the forced and free convective flow of an electrically conducting fluid past a vertical semi-infinite plate, on taking into account dissipative heat and stress work, have been presented. Without magnetic field, it has been discussed by the authors [1] in an earlier paper. The effects of Gr (Grashof number, Gr>0 cooling of the plate by free convection currents, Gr<0 heating of the plate by free convection currents), Pr (Prandtl number), F (Froude number) and M² (the magnetic field parameter) are discussed. It is observed that reverse type of flow of air exists near the plate when Gr<0.

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Gemischte erzwungene und freie Konvektions-MHD-Strömung an einer halbunendlichen senkrechten Platte

Zusammenfassung Die Wirkung eines transversalen Magnetfeldes auf die erzwungene und freie Konvektion einer elektrisch leitenden Flüssigkeit an einer halbunendlichen senkrechten Platte wurde unter Berücksichtigung der Dissipationswärme und der Kompressionsarbeit mitgeteilt. Das Problem wurde ohne Magnetfeld schon früher [1] behandelt. Diskutiert wurde die Wirkung der Grashof-Zahl Gr (Gr>0 Kühlung der Platte durch freie Konvektion), der Prandtl-Zahl Pr, der Froude-Zahl F und des magnetischen Feldparameters M₂. Bei Gr<0 wird Umkehr strömung in Plattennähe betrachtet.