Studies on gravitational spreading currents

Gas-liquid two-phase turbulent flow along a vertical flat plate immersed in bubble flow was numerically simulated using the SIMPLER method. Prandtl's mixing length theory was applied to estimate eddy viscosity. The predicted turbulent velocity profiles were in good agreement with the measurements in a bubble flow near the vertical plate, compared with the calculated laminar velocity profiles.

The resistance law for flow on river-beds with vegetation is investigated experimentally and theoretically. The results of basic experiments with artificial vegetation indicate that when the vegetation density is less than a certain level, the flow passing through the vegetation is considerably affected by the river-bed condition. A numerical model for the vertical velocity distribution and the mean flow velocity is derived, based on Reynolds stress reduction model and Prandtl's mixinglength theory. The validity of this model is demonstrated through its comparison with the experiments conducted by the authors.

The various aspects of the propagation of long waves onto a shelf (i.e., reflection, transmission and propagation on the shelf) are examined experimentally and theoretically. The results are applied to tsunamis propagating onto the continental shelf. A numerical method of solving the one-dimensional Boussinesq equations for constant depth using finite element techniques is presented. The method is extended to the case of an arbitrary variation in depth (i.e., gradually to abruptly varying depth) in the direction of wave propagation. The scheme is applied to the propagation of solitary waves over a slope onto a shelf and is confirmed by experiments. A theory is developed for the generation in the laboratory of long waves of permanent form, i.e., solitary and cnoidal waves. The theory, which incorporates the nonlinear aspects of the problem, applies to wave generators which consist of a vertical plate which moves horizontally. Experiments have been conducted and the results agree well with the generation theory. In addition, these results are used to compare the shape, celerity and damping characteristics of the generated waves with the long wave theories. The solution of the linear nondispersive theory for harmonic waves of a single frequency propagating over a slope onto a shelf is extended to the case of solitary waves. Comparisons of this analysis with the nonlinear dispersive theory and experiments are presented. Comparisons of experiments with solitary and cnoidal waves with the predictions of the various theories indicate that, apart from propagation, the reflection of waves from a change in depth is a linear process except in extreme cases. However, the transmission and the propagation of both the transmitted and the reflected waves in general are nonlinear processes. Exceptions are waves with heights which are very small compared to the depth. For these waves, the entire process of propagation onto a shelf in the vicinity of the shelf is linear . Tsunamis propagating from the deep ocean onto the continental shelf probably fall in this class.

R wave is short for Rayleigh wave. R wave,on which the effect of gravity is generally inevitable,has received considerably extensive attention and been widely used in many fields,such as geological exploration,nondestructive test,etc. The effects of gravity on R wave velocity for elastic parameters were further analyzed based on available R wave velocity function,and the applicable range of the approximate one was given as well. Wave propagation through quasi-half-space isotropic linear elastic medium with gravity,under the excitation of differentiated Gaussian pulse( DGP) was studied by finite difference time domain( FDTD) method with staggered grid discretization,and extended boundary conditions were introduced.Velocity values obtained were closer to the theoretical ones. Meanwhile,the influence of gravity on time domain,frequency domain response was discussed. By analyzing the theoretical and numerical results,it is necessary to take the effect of gravity into account in simulation for more accurate prediction.

A domain of influence theorem is proved for a linear thermoelastic solid with a Cattaneo's type heat conduction law and a scalar internal variable. The obtained result is applied to prove the hyperbolicity of a semiempirical heat conduction theory, describing the propagation of thermal waves in crystals at low temperatures.

This paper is concerned with the linear signal response analysis of a thin viscous liquid sheet which is at rest in an appropriate frame of reference and in contact with passive external media, and on which localized external pressures act from the passive media as sources of perturbation. The frame of the analysis is provided by general formulae for the response signals of the sheet in the two excitation modes, sinuous and varicose, which result as the solution of the appropriate fluid dynamic initial-boundary value problems by the Fourier-Laplace transform technique. These formulae display how the signals depend on the nature of the perturbation and on the spectrum of the (linear) eigenmodes of the sheet. The signals can be evaluated either numerically or, as initiated in this paper, analytically, in long wave approximations. The long wave approximation will be seen in the sequel to concentrate on particular eigenmodes of the sheet spectrum, with small values k of the wave vector % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabm4Aayaala % aaaa!36F0! $${\vec k}$$ along the sheet. The present paper is devoted mainly to a detailed analysis of the spectrum of eigenmodes of the sheet, and to the formulation of long wave approximations of (linear) response signals of the sheet in the context of this analysis. It turns out that the sheet spectrum of eigenmodes depends on only one characteristic number Γ, which depends on the (positive) fluid parameters: the surface tension γ at (both) the interfaces of the sheet, the density ρ of the fluid, its kinematic viscosity ν, and on the thickness h of the sheet: % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaey % ypa0ZaaOaaaeaadaWcaaqaaiabeo7aNjaadIgaaeaacaaIYaGaeqyW % diNaeqyVd42aaWbaaSqabeaacaaIYaaaaaaaaeqaaOGaaiOlaaaa!40E9! $$\Gamma = \sqrt {\frac{{\gamma h}} {{2\rho \nu ^2 }}} .$$ This number is closely related, by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdCKaey % ypa0ZaaSaaaeaacaaIXaaabaWaaOaaaeaacaaIYaaaleqaaaaakiaa % d+eacaWGObWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaiilaaaa!3E58! $$\Gamma = \frac{1} {{\sqrt 2 }}Oh^{ - 1} ,$$ to the Ohnesorge number % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4taiaadI % gacqGH9aqpdaWcaaqaaiabeY7aTbqaamaakaaabaGaeq4SdCMaeqyW % diNaamiAaaWcbeaaaaGccaGGSaaaaa!3FA4! $$Oh = \frac{\mu } {{\sqrt {\gamma \rho h} }},$$ where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiVd0Maey % ypa0JaeqyWdiNaeqyVd4gaaa!3C22! $$\mu = \rho \nu $$ is the dynamic viscosity of the liquid. It will be shown in the sequel that of the infinitely many branches of the sheet spectrum only two (pairs of) ’soft’ branches, one sinuous and one varicose, will be relevant for a long wave approximation. For these branches asymptotic expansions of the dispersion relations ω(k) between the (complex) mode frequencies ω and % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 % da9iaacYhaceWGRbGbaSaacaGG8baaaa!3AE6! $$k = |\vec k|$$ for long waves (i.e. for k → 0), which obey ω(k) → 0 with k → 0, will be derived. The analytic long wave dispersion relations for the (soft) sinuous and varicose spectral branches allow a very favourable insight into the qualitative mode behaviour, including analogies beyond liquid sheets. They show e.g. that to their lowest orders in k the soft sinuous modes are nondispersive, i.e. % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHeSaeq % yYdCNaeyyhIuRaam4AaiaacYcaaaa!3C5A! $$\Im \omega \propto k,$$ and only weakly absorptive, i.e. ℜ ω ∝ k2 for low viscosity and ℜω ∝ k4 for high viscosity (ℜω < 0). Their signals are therefore expected to have some resemblance to signals of a flexible membrane on the one side and those of a (2-dimensional) diffusion process, or a hyperdiffusion process where % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Iaey % 4bIe9aaWbaaSqabeaacaqGYaaaaaaa!3943! $$ - \nabla ^{\text{2}} $$ is replaced by % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaW % baaSqabeaacaaI0aaaaOGaaiOlaaaa!391B! $$\nabla ^4 .$$ The behaviour of the soft varicose modes will be seen to depend on the value of Γ: For Γ2 < 4 the asymptotic expansion for k → 0 gives $$ \Im \omega = {\text{0}} $$ and ℜω ∝ k2(ℜω 4 the situation is different: here % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX! % MathType!MTEF!2!1!+- % feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHeSaeq % yYdCNaeyyhIuRaam4AamaaCaaaleqabaGaaGOmaaaaaaa!3C93! $$\Im \omega \propto k^2 $$ and ℜω ∝ k2 (ℜω < 0). This behaviour bears resemblance to the vibration modes of a thin elastic plate with (diffusion-like) damping. The asymptotic insight into the behaviour of the individual branches of spectral modes allows the derivation of rather transparent approximate analytic expressions for the response signals of the sheet to external perturbations.

Dilute polymer solutions of sufficiently high molecular weight generally exhibit the phenomenon of drag reduction in turbulent flows. This has been usually studied by conducting either pressure drop/flow rate measurements or axial velocity profile measurements. A general correlation, based on Prandtl's mixing length model of the form, is developed herein to correlate the velocity data obtained in a 9 mm glass tube using an LDA system. This new correlation can be used to describe the velocity data of dilute solutions of polyacrylamide, xanthan gum and polyethylene oxide. The parameters α, β and Γ are unique functions of u */〈ū〉 and are independent of polymer type, concentration, mechanical degradation and Reynolds number. Good agreement between two independent data sets and predictions of the velocity profile using the above correlation was observed.

Linear theory is developed of flexural wave motions of an elastic beam of circular cross-section traveling along its axial direction at a constant speed in an air-filled, circular tube placed coaxially. The beam is constrained to deflect in a plane and is subjected to a restoring force in proportion to the magnitude of deflection. Both beam and tube are assumed to be long enough for end effects to be ignored. Taking account of aerodynamic loading on the lateral surface of the beam, wave propagation is examined under a long-wave approximation that a characteristic wavelength of flexural waves is much longer than the tube's radius. Using the lossless, acoustic wave equation for the velocity potential of the air, and the classical, flexural wave equation for the deflection of the beam, an asymptotic-expansion method is applied based on the long-wave approximation. A linear wave equation for the deflection is derived and its dispersion relation is examined by comparing with the exact one derived previously without any long-wave approximation. Higher-order effects of the axial curvature of the beam and the compressibility of air on the aerodynamic loading are also discussed.

Gravity currents are complex phenomena occurring in a wide range of natural flows. They were simulated in this research with the TELEMAC3D modelling system. The numerical results were compared to two sets of laboratory scale 2D and 3D lock–exchange experiments. In the 2D case, numerical and experimental results agreed very well. While in the 3D case a fair agreement was obtained. It was essential to use the non–hydrostatic approach and suitable turbulence models (Prandtl's mixing length and the k–e models). We conclude that 3D non–hydrostatic numerical model is capable of simulating gravity currents in regular geometry. The two lock–exchange experiments can serve as benchmark cases for realistic density currents in fluvial or estuarine environment.

: An integral method is presented for estimating the influence of an initial boundary layer on the development of a two-dimensional, isobaric, turbulent, free shear layer. The basic equation is derived by applying the principle that, at any streamwise station along the free shear layer, the momentum of the entrained flow equals the total axial turbulent shear force acting along the dividing streamline. This equation is solved using a single parameter family of velocity profiles derived by Korst and Prandtl's mixing length concept for turbulent shear stress. The theory involves one empirical constant which was evaluated using Tollmien's experimental data for incompressible, turbulent mixing. The theory is verified by comparing with experimental data for free-stream Mach numbers up to 6.4.

A new model for mean velocity profile of turbulent water flow with added drag-reducing surfactants is presented in this paper. The general problem of drag due to frictional resistance is reviewed and drag reduction by the addition of polymers or surfactants is introduced. The model bases on modified Prandtl's mixing length hypothesis and includes three parameters, which depend on additives and can be evaluated by numerical simulation from experimental datasets. The advantage of the model in comparison with previously reported models is that it gives uniform curve for whole pipe section and can be determined for a new surfactant with less necessary measurements. The use of the model is demonstrated for surfactant Habon-G as an example.

We present a new model of the drag force generated by bluff bodies accelerating in a static fluid, and also accelerating in a headwind or decelerating in a tailwind, while sustaining an external force proportional to acceleration. Specific equations for the evolution of the drag coefficient are derived from the disk tow tank data of I versen and Balent ( Journal of Applied Physics , 22 , pp. 324 – 328, 1951). Expressions for the drag force caused by motions in a tailwind and in a headwind are then derived using Galilean transformations of the tow tank data, which involved a static fluid en vironment far ahead of the moving disk. Application of these results to powered disk -like bodies are discussed. The validity of applying disk data to other bluff bodies is shown to be strongly dependent on the acceleration modulus. Prandtl's mixing length hypothesis is used to derive a useful formula from which this assessment is made. Finally, the issue of dynamical similarity for turbulent flows is revisited and extended to account for the unsteadiness of the flow that exists ahead of the disks. Nomencla ture

Ludwig Prandtl and the growth of fluid mechanics in Germany

Formation of large structure in the Universe as a result of gravitational instability in cold dark matter is investigated in a simple analytical model. Collapse of the rotating spheroid is approximated by a system of ordinary differential equations describing its dynamics. The gravitational potential is approximated by the one of the uniform Maclaurin spheroid. Development of gravitational instability and collapse in tlie dark matter medium do not lead to any shock formation or radiation, but is characterized by non-colliskmal relaxation, which is accompanied by the mass and angular momentum losses. Phenomenological account of these processes is done in this model. Formation of the equilibrium configuration dynamics of collapse is investigated. A very long gravitational wave emission during the collapse is estimated, and their possible connection with the observed gravitational lenses is discussed.

To study the energy dissipation mechanism of nonobstructive particle dampers (NOPDs) and provide guidance to the application of NOPDs, the dense granular flow theory was introduced to establish a quantitative energy dissipation model for NOPDs. The convection movement of the particles under vibrational excitations was studied using the discrete element method, and the Prandtl mixing length theory was adopted to modify the constitution law of dense granular flows. The pressure of the granular flow was obtained by equivalenting the vibrational excitation to a body force acted on particles. Theoretical results showed that the energy dissipation rate of the NOPD was increased with the vibration intensity and decreased with the granular diameter. It also indicated that particles near the side wall and the bottom of the damper dissipated more energy than those particles in other regions. The theoretical model was verified by simulation and experimental result. The results may provide a new approach to studying the energy dissipation mechanism of NOPD and give some guidance to enhancing the damping performance of NOPD in engineering practices.