Resonant generation of a solitary wave in a thermocline

The resonant generation of a second-mode internal solitary wave, resulting from a ship internal waves system damping in a thermocline, is studied experimentally. The source of the stationary internal waves was provided by an oblong ellipsoid of revolution towed horizontally and uniformly at the depth of the thermocline center. The ranges of the Reynolds and Froude numbers were 500Re=Ul/v 15000 and 0.3Fi=U/N _max D1.0, respectively. When the body's speed and the linear long-wave second-mode phase speed were equal, an internal solitary wave of the bulge type was observed. The shape of the wave satisfied the Korteweg-de Vries equation. The Urcell parameter was equal to 10.2.List of Symbols L, B, H towing tank length, breadth and height respectively - z vertical coordinate - D characteristic vertical dimension of the body - a minor semiaxis of an ellipsoid - b major semiaxis of an ellipsoid (maximum ellipsoid diameter D=2a) - l length of the body ( =2b) - U velocity of the body - t temperature - g acceleration due to gravity - i fresh water density at ith level - _4° fresh water density for temperature t=4°C - o water density at the center of the thermocline - _i density variation due to the temperature variation at the ith horizon - N Brunt-Väisälä frequency - N _max maximum value of Brunt-Väisälä frequency - Re Reynolds number - Fi internal Froude number - f _n eigenfunction of the boundary-value problem for the nth mode - _n nth mode frequency - k _{n n}th mode horizontal wavenumber - C _n limiting phase speed of a linear nth mode interval wave (= _n/k_n;k_n 0) - Ur Urcell parameter - v fresh water kinematic viscosity - conventional density - half-length of a solitary wave - ₀ solitary wave height - time This work was partially supported by the INTAS (grant no. 94-4057) and by the Russian Foundation of Basic Research under grant no. 94-05-17004-a.A version of this paper was presented at the Second International Conference on Experimental Fluid Mechanics, Torino, Italy, 4–8 July, 1994.     

Critical phenomena and waves in gas-liquid systems

The results of the hydraulic studies of gas-liquid media, wave processes in two-phase media and critical phenomena are described. Some methodological foundations to describe these media and methods to obtain the basic similarity criteria for the hydraulics and gas-dynamics of bubble suspensions are discussed. A detailed consideration is given for the phase transition processes on interfaces and the interface stability. A relation has been revealed between the wave and critical phenomena in two-phase systems.Nomenclature a thermal diffusivity - Ar Archimedes number - B gas constant - C heat capacity - C _p heat capacity at constant pressure - C _v heat capacity at constant volume - c ₀ acoustic velocity in the mixture - c _l acoustic velocity in the liquid - C _f flow resistance coefficient - G mass rate of flow - g gravitational acceleration - L latent heat of evaporation - l initial perturbation width - M Mach number - Nu Nusselt number - P pressure - Pr Prandtl number - R bubble radius - (3P ₀/R ₀ ² _f )^–1 bubble resonance frequency square - T temperature - U medium motion velocity - W heavy phase velocity - W light phase velocity - We Weber number - heat release coefficient - dispersion coefficient - void fraction - adiabatic index - film thickness - dimensionless film thickness - kinematic viscosity coefficient - dynamical viscosity coefficient - dissipation coefficient in the mixture - dispersion parameter - _f liquid phase density - light phase density - heat conductivity - surface tension - frequency, ₀ ² =3P ₀/ _f R ₀ ²     

Lp-estimates for the nonlinear spatially homogeneous Boltzmann equation

This paper studies L^p-estimates for solutions of the nonlinear, spatially homogeneous Boltzmann equation. The molecular forces considered include inverse k^th-power forces with k > 5 and angular cut-off.The main conclusions are the following. Let f be the unique solution of the Boltzmann equation with f(v,t)(1 + ¦v²¦)^(s ₁ ^{+ /p)/2} L¹, when the initial value f ₀ satisfies f ₀(v) 0, f ₀(v) (1 + ¦v¦²)^(s ₁ ^{+ /p)/2} L¹, for some s₁ 2 + /p, and f ₀(v) (1 + ¦v¦²)^s/2 L^p. If s 2/p and 1 < p < , then f(v, t)(1 + ¦v¦²)^{(s s} ₁)^/2 L^p, t > 0. If s >2 and 3/(1+ ) < p < , thenf(v,t) (1 + ¦v¦²)^(s(s ₁ ^{+ 3/p))/2} L^p, t > 0. If s >2 + 2C₀/C₁ and 3/(l + ) < p < , then f(v,t)(1 + ¦v¦²)^s/2 L^p, t > 0. Here 1/p + 1/p = 1, x y = min (x, y), and C₀, C₁, 0 < 1, are positive constants related to the molecular forces under consideration; = (k – 5)/ (k – 1) for k^th-power forces.Some weaker conclusions follow when 1 < p 3/ (1 + ).In the proofs some previously known L-estimates are extended. The results for L^p, 1 < p < , are based on these L-estimates coupled with nonlinear interpolation.     

Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Deformation to breaking of deep water gravity waves

The results of laboratory observations of the deformation of deep water gravity waves leading to wave breaking are reported. The specially developed visualization technique which was used is described. A preliminary analysis of the results has led to similar conclusions than recently developed theories. As a main fact, the observed wave breaking appears as the result of, first, a modulational instability which causes the local wave steepness to approach a maximum and, second, a rapidly growing instability leading directly to the breaking.List of symbols L total wave length - H total wave height - crest elevation above still water level - trough depression below still water level - wave steepness =H/L - crest steepness =/L - trough steepness =/L - F ₁ forward horizontal length from zero-upcross point (A) to wave crest - F ₂ backward horizontal length from wave crest to zero-downcross point (B) - crest front steepness =/F ₁ - crest rear steepness =/F ₂ - vertical asymmetry factor=F ₂/F ₁ (describing the wave asymmetry with respect to a vertical axis through the wave crest) - µ horizontal asymmetry factor=/H (describing the wave asymmetry with respect to a horizontal axis: SWL) - T ₀ wavemaker period - L ₀ theoretical wave length of a small amplitude sinusoïdal wave generated at T _inf0 ^sup–1 frequency - ₀ average wave height     

Flow in porous media II: The governing equations for immiscible,two-phase flow

The Stokes flow of two immiscible fluids through a rigid porous medium is analyzed using the method of volume averaging. The volume-averaged momentum equations, in terms of averaged quantities and spatial deviations, are identical in form to that obtained for single phase flow; however, the solution of the closure problem gives rise to additional terms not found in the traditional treatment of two-phase flow. Qualitative arguments suggest that the nontraditional terms may be important when / is of order one, and order of magnitude analysis indicates that they may be significant in terms of the motion of a fluid at very low volume fractions. The theory contains features that could give rise to hysteresis effects, but in the present form it is restricted to static contact line phenomena.Roman Letters (, = , , and ) A interfacial area of the- interface contained within the macroscopic system, m² - A _e area of entrances and exits for the -phase contained within the macroscopic system, m² - A interfacial area of the- interface contained within the averaging volume, m² - A ^* interfacial area of the- interface contained within a unit cell, m² - A _e ^* area of entrances and exits for the-phase contained within a unit cell, m² - g gravity vector, m²/s - H mean curvature of the- interface, m^–1 - H area average of the mean curvature, m^–1 - H – H , deviation of the mean curvature, m^–1 - I unit tensor - K Darcy's law permeability tensor, m² - K permeability tensor for the-phase, m² - K viscous drag tensor for the-phase equation of motion - K viscous drag tensor for the-phase equation of motion - L characteristic length scale for volume averaged quantities, m - characteristic length scale for the-phase, m - n unit normal vector pointing from the-phase toward the-phase (n = –n ) - p _c p – P , capillary pressure, N/m² - p pressure in the-phase, N/m² - p intrinsic phase average pressure for the-phase, N/m² - p – p , spatial deviation of the pressure in the-phase, N/m² - r ₀ radius of the averaging volume, m - t time, s - v velocity vector for the-phase, m/s - v phase average velocity vector for the-phase, m/s - v intrinsic phase average velocity vector for the-phase, m/s - v – v , spatial deviation of the velocity vector for the-phase, m/s - V averaging volume, m³ - V volume of the-phase contained within the averaging volume, m³ Greek Letters V /V, volume fraction of the-phase - mass density of the-phase, kg/m³ - viscosity of the-phase, Nt/m² - surface tension of the- interface, N/m - viscous stress tensor for the-phase, N/m² - / kinematic viscosity, m²/s     

A phase plane discussion of convergence to travelling fronts for nonlinear diffusion

The paper is concerned with the asymptotic behavior as t of solutions u(x,t) of the equation in the case f(0)=f(1)=0, with f(u) non-positive for u(>0) sufficiently close to zero and f(u) non-negative for u(<1) sufficiently close to 1. This guarantees the uniqueness (but not the existence) of a travelling front solution u;U(x–ct), U(–);0, U();, and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front, if it exists, and to a stacked combination of travelling fronts if it does not. The approach is to use the monotonicity to take u and t as independent variables and p = u _x as the dependent variable, and to apply ideas of sub- and super-solutions to the diffusion equation for p.This research was sponsored by the United States Army under Contract No. DAAG29-75-C-0024.     

Similar solutions and the asymptotics of filtration equations

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t. We prove that similar solutions of the equation u _t = (u )_xx asymptotically represent solutions of the Cauchy problem for the full equation u _t = [(u)]_xx if (u) is close to u for small u.     

Effect of measuring volume length on two-component laser velocimeter measurements in a turbulent boundary layer

The effects of finite measuring volume length on laser velocimetry measurements of turbulent boundary layers were studied. Four different effective measuring volume lengths, ranging in spanwise extent from 7 to 44 viscous units, were used in a low Reynolds number (Re=1440) turbulent boundary layer with high data density. Reynolds shear stress profiles in the near-wall region show that u v strongly depends on the measuring volume length; at a given y-position, u v decreases with increasing measuring volume length. This dependence was attributed to simultaneous validations on the U and V channels of Doppler bursts coming from different particles within the measuring volume. Moments of the streamwise velocity showed a slight dependence on measuring volume length, indicating that spatial averaging effects well known for hot-films and hot-wires can occur in laser velocimetry measurements when the data density is high.List of symbols time-averaged quantity - u wall friction velocity, ( _w /)^1/2 - v kinematic viscosity - d _p pinhole diameter - l _eff spanwise extent of LDV measuring volume viewed by photomultiplier - l ⁺ non-dimensional length of measuring volume, l _eff u /v - y ⁺ non-dimensional coordinate in spanwise direction, y u /v - z ⁺ non-dimensional coordinate in spanwise direction, z u /v - U ⁺ non-dimensional mean velocity, /u - u instantaneous streamwise velocity fluctuation, U &#x2329;U - v instantaneous normal velocity fluctuation, V–V - u RMS streamwise velocity fluctuation, u ^21/2 - v RMS normal velocity fluctuation, v ^21/2 - Re Reynolds number based on momentum thickness, U 0/v - R _uv cross-correlation coefficient, u v/u v - R₁₂(0, 0, z) two point correlation between u and v with z-separation, <u(0, 0, 0) v (0, 0, z)>/<u(0, 0, 0) v (0, 0, 0)> - N rate at which bursts are validated by counter processor - T Taylor time microscale, u (dv/dt²)^–1/2     

The dynamic performance of the Weissenberg rheogoniometer III. Large amplitude oscillatory shearing — harmonic analysis

The harmonic content of the nonlinear dynamic behaviour of 1% polyacrylamide in 50% glycerol/water was studied using a standard Model R 18 Weissenberg Rheogoniometer. The Fourier analysis of the Oscillation Input and Torsion Head motions was performed using a Digital Transfer Function Analyser.In the absence of fluid inertia effects and when the amplitude of the (fundamental) Oscillation Input motion _I is much greater than the amplitudes of the Fourier components of the Torsion Head motion _Tn empirical nonlinear dynamic rheological propertiesG _n (, ⁰),G _n (, ⁰) and/or _n (, ⁰), _n (, ⁰) may be evaluated without a-priori-knowledge of a rheological constitutive equation. A detailed derivation of the basic equations involved is presented.Cone and plate data for the third harmonic storage modulus (dynamic rigidity)G ₃ (, ⁰), loss modulusG ₃ (, ⁰) and loss angle ₃ (, ⁰) are presented for the frequency range 3.14 × 10^–2 1.25 × 10² rad/s at two strain amplitudes, _CP ⁰ = 2.27 and 4.03. Composite cone and plate and parallel plates data for both the third and fifth harmonic dynamic viscosities ₃ (, ⁰), _S (, ⁰) and dynamic rigiditiesG ₃ (, ⁰),G ₅ (, ⁰) are presented for strain amplitudes in the ranges 1.10 _CP ⁰ 4.03 and 1.80 _PP ⁰ 36 for a single frequency, = 3.14 × 10^–1 rad/s. Good agreement was obtained between the results from both geometries and the absence of significant fluid inertia effects was confirmed by the superposition of the data for different gap widths.     

The behavior of a spherical bubble near a solid wall in a compressible liquid

Summary The behavior of a spherical bubble near a solid wall is analysed by considering the liquid compressibility. The equation of motion of the bubble with first order correction for the effects of liquid compressibility and solid wall is derived. The equation obtained here coincides with the known result in case of L or C . Further experimental study is made on the motion of bubbles produced by a spark discharge in water. The theoretical results are in good agreement with the experiments.

---

Das Verhalten einer kugelförmigen Blase in einer kompressiblen Flüssigkeit in der Nähe einer festen Wand

Übersicht Bei Berücksichtigung der Flüssigkeitskompressibilität wird das Verhalten einer kugelförmigen Blase in der Nähe einer festen Wand analysiert. Die Gleichung der Bewegung der Blase wird mit der Korrektur erster Ordnung für den Einfluß der Flüssigkeitskompressibilität und der festen Wand angegeben. Aus der erhaltenen Gleichung wird für L oder C das bekannte Ergebnis hergeleitet. Darüber hinaus wird eine experimentelle Untersuchung der Blasenbewegung durchgeführt. Die Blase wird mit Hilfe von Funkendurchschlägen zwischen Elektroden in Wasser erzeugt. Die theoretischen Ergebnisse stimmen gut mit den Experimenten überein.

     

Flow visualization of compressible vortex structures using density gradient techniques

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.Nomenclature a Unobstructed height of schlieren light source in cutoff plane, m - c Blade chord, m - f Focal length of schlieren focusing mirror, m - C _T Rotor thrust coefficient, T/( ² R ⁴) - I Image screen illumination, Lm/m ² - l Distance from vortex to shadowgraph screen, m - n _b Number of blades - p Pressure,N/m ² - p Ambient pressure, N/m ² - r, , z Cylindrical coordinate system - r _c Vortex core radius, m - Non-dimensional radial coordinate, (r/r _c ) - R Rotor radius, m - Tangential velocity, m/s - Specific heat ratio of air - Circulation (strength of vortex), m ²/s - Non-dimensional quantity, ² 8²p r _c ² - Refractive index of fluid medium - ₀ Refractive index of fluid medium at reference conditions - Gladstone-Dale constant, m ³/kg - Density, kg/m ³ - Density at ambient conditions, kg/m ³ - Non-dimensional density, (/ ) - Rotor solidity, (n _b c/ R) - Rotor rotational frequency, rad/s     

Existence theorem for a minimum problem with free discontinuity set

We study the variational problem Where is an open set in ⁿ ,n2gL ^q () L (), 1q<+, O<, <+ andH _n–1 is the (n–1)-dimensional Hausdorff Measure.     

Torsion of two bonded layers by a rigid disk

The problem considered in this paper describes the torsion of a homogeneous isotropic elastic layer (0zd ₁) of finite thickness d ₁, perfectly bonded to another elastic layer (-d ₂z0) of finite thickness d ₂. The problem is reduced to the solution of a Fredholm integral equation of the second kind. The solutions are given for some particular cases.

---

Sommario In questo lavoro si considera il problema della torsione di uno strato elastico omogeneo ed isotropo (0zd ₁) di spessore finito d ₁, perfettamente incollato ad un altro strato elastico (-d ₂z0) di spessore finito d ₂. II problema é ricondotto alla soluzione di una equazione integrale di Freedholm del secondo ordine. Le soluzioni sono ottenute per alcuni casi particolari.

     

Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions

In this work we consider transport in ordered and disordered porous media using singlephase flow in rigid porous mediaas an example. We defineorder anddisorder in terms of geometrical integrals that arise naturally in the method of volume averaging, and we show that dependent variables for ordered media must generally be defined in terms of thecellular average. The cellular average can be constructed by means of a weighting function, thus transport processes in both ordered and disordered media can be treated with a single theory based on weighted averages. Part I provides some basic ideas associated with ordered and disordered media, weighted averages, and the theory of distributions. In Part II a generalized averaging procedure is presented and in Part III the closure problem is developed and the theory is compared with experiment. Parts IV and V provide some geometrical results for computer generated porous media.Roman Letters A interfacial area of the- interface contained within the macroscopic region, m² - A_e area of entrances and exits for the-phase contained within the macroscopic system, m² - g gravity vector, m/s² - I unit tensor - K traditional Darcy's law permeability tensor, m² - L general characteristic length for volume averaged quantities, m - characteristic length (pore scale) for the-phase - (y) weighting function - m(–y) (y), convolution product weighting function - _v special weighting function associated with the traditional averaging volume - N unit normal vector pointing from the-phase toward the-phase - p pressure in the-phase, N/m² - p0 reference pressure in the-phase, N/m² - p traditional intrinsic volume averaged pressure, N/m² - r0 radius of a spherical averaging volume, m - r position vector, m - r position vector locating points in the-phase, m - averaging volume, m³ - V volume of the-phase contained in the averaging volume, m³ - V _cell volume of a unit cell, m³ - v velocity vector in the-phase, m/s - v traditional superficial volume averaged velocity, m/s - x position vector locating the centroid of the averaging volume or the convolution product weighting function, m - y position vector relative to the centroid, m - y position vector locating points in the-phase relative to the centroid, m Greek Letters indicator function for the-phase - Dirac distribution associated with the- interface - V/V, volume average porosity - mass density of the-phase, kg/m³ - viscosity of the-phase, Ns/m²     

Convergence for Semilinear Degenerate Parabolic Equations in Several Space Dimensions

We show that any global nonnegative and bounded solution to the degenerate parabolic problemu_t-u^m+f(u)=0 qquad {\rm on} quad R^N,u|{}=0converges to a single stationary state as time goes to infinity. Here m>0, f is a restriction of a real analytic function defined on a sector containing the half-line [0, ), and f(u ^1/m ) is a continuously differentiable function of u.     

Some aspects of dilute suspension theory

A theory proposed by the author as representative of the flow of a general suspension contains three interaction forces, f, S and N. For a quasi-concentrated suspension and for a dilute suspension, N and S, N are omitted, respectively. For the latter special case, we treat diffusion of a fluid through an elastic solid. For a quasi-concentrated suspension, we show that F and S depend on the gradient of the motion gradient. We demonstrate the existence of interesting phenomena: non-simple behavior, dissipative effects, generalized lift and drag forces.Presented at the second conference Recent Developments in Structured Continua, May 23 – 25, 1990, in Sherbrooke, Québec, Canada.     

A similarity law for acute circular cones at large angles of attack in a supersonic flow

For thin bodies placed in a hypersonic flow at a small angle of attack the similarity law is known. From this law it follows that for various numbers M, angles of attack , and relative thicknesses the similarity conditions will be observed if in the flows under consideration the parameters M and / are the same. This similarity law is obtained with the assumption M 1, 1. But even for M=3 and 1/3 the results of solving the complete system of gasdynamic equations for affino-similar bodies is in a good agreement with the similarity law [1], In [2] it is shown that this similarity law is generalized for the case of a flow around a thin pointed body at large angles of attack. According to the similarity law, at large angles of attack the flows near bodies with an identical distribution of cross-sectional shapes will be similar if the parameters K₁= cotan and K₂=m sin for all cases have one and the same value. As the angle of attack decreases, the requirements of constancy of K₁ and K₂ become analogous to the conditions M=const, /=const.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 78–83, May–June, 1976.The authors thank V. V. Lunev for the useful discussions and valuable observations.     

On shock layer method for the hypersonic flow problems

Chernyi’s series method[1] is not proper for the case that(γ-l)/(γ+l)<<2/(γ+1)×M²sin²β (γ=c_p/c_v-adiabatic index number, M-Much number, β-shock incidence). In this paper, we only suppose that in the neighbour of the shock, there exists a shock layer in which the density of the gas is very big, but we do not remove the case that (γ-1)/(γ+1)<<2/(γ+1)M²sin²β.     

Laser velocimetry measurements in a horizontal gas-solid pipe flow

This paper presents laser measurements of particle velocities in a horizontal turbulent two-phase pipe flow. A phase Doppler particle analyzer, (PDPA), was used to obtain particle size, velocity, and rms values of velocity fluctuations. The particulate phase consisted of glass spheres 50 m in diameter with the volume fraction of the suspension in the range _p=10^-4 to _p=10^-3. The results show that the turbulence increases with particle loading.List of symbols a particle diameter - C _va velocity diameter cross-correlation - d pipe diameter - Fr ² Froude number - g gravitational constant - p(a) Probability density of the particle diameter - Re pipe Reynolds number based on the friction velocity - T characteristic time scale of the energy containing eddies - T _L integral scale of the turbulence sampled along the particle path - u, U, u characteristic fluid velocities: fluctuating, mean and friction - v characteristic velocity of the paricle fluctuations - f expected value of any random variable f - f¦g expected value of f given a value of the random variable g - _p particle volume fraction - _p particle response time - absolute fluid viscosity - v kinematic fluid viscosity - _p, _f densities, particle and fluid - _a ² particle diameter variance - _va ² velocity variance due to the particle diameter variance - _vT ² total particle velocity variance - _vt ² particle velocity variance due to the response to the turbulent field