Three-dimensional non-hydrostatic simulation of gravity currents using TELEMAC3D and comparison of results to experimental data

Ludwig Prandtl and the growth of fluid mechanics in Germany

Shear-induced vertical mixing in a stratified flow is a key ingredient of thermohaline circulation. We experimentally determine the vertical flux of momentum and density of a forced gravity current using high-resolution velocity and density measurements. A constant eddy-viscosity model provides a poor description of the physics of mixing, but a Prandtl mixing length model relating momentum and density fluxes to mean velocity and density gradients works well. For the average gradient Richardson number Ri(g) approximately 0.08 and a Taylor Reynolds number Re(lambda) approximately 100, the mixing lengths are fairly constant, about the same magnitude, comparable to the turbulent shear length.

A fundamental study was conducted to shed light on entrainment and mixing in buoyancy-driven Boussinesq density currents. Large-eddy simulation was performed on lock-exchange (LE) release density currents—an idealized test bed to generate density currents. As dense fluid was released over a sloping surface into an ambient lighter fluid, the dense fluid slumps to the bottom and forms a characteristic head of the current. The dynamics of the head dictated the mixing processes in LE currents. The key contribution of this study is to resolve an ongoing debate on mixing: We demonstrate that substantial mixing occurs in the early stages of evolution in an LE experiment and that entrainment is highly inhomogeneous and unsteady during the slumping regime. Guided by the flow physics, entrainment is calculated using two different but related perspectives. In the first approach, the entrainment parameter (E) is defined as the fraction of ambient fluid displaced by the head that entrains into the current. It is an indicator of the efficiency in which ambient fluid is displaced into the current and it serves as an important metric to compare the entrainment of dense currents over different types of surfaces, e.g., roughness configuration. In the second approach, E measures the net entrainment in the current at an instantaneous time t over the length of the current. Net entrainment coefficient is a metric to compare the effects of flow dynamical conditions, i.e., lock-aspect ratio that dictates the fraction of buoyancy entering the head, and also the effect of the sloping angle. Together, the entrainment coefficient and the net entrainment coefficient provide an insight into the entrainment process. The “active” head of the current acts as an engine that mixes the ambient fluid with the existing dense fluid, the 3-D lobes and clefts on the frontal end of the current causes recirculation of the ambient fluid into the current, and Kelvin-Helmholtz rolls are the mixers that entrain the ambience into the current. Buoyancy and shear production occur at the interface in the head region of the current, and transport of turbulence kinetic energy (TKE) by Reynolds stresses results in high TKE.

Field observations imply that accumulation and advection of invertebrate larvae and other plankton by propagating fronts provides an efficient mechanism for the cross‐shelf transport and subsequent coastal settlement of larvae and plankton. Gravity currents propagating into a heavier, uniform‐density fluid are known to have near‐surface velocities u that exceed the propagation speed c of the gravity current nose. It is shown by simple dam‐break laboratory experiments and through corresponding numerical solutions of two‐dimensional stratified flow and Lagrangian particle advection that this characteristic of gravity currents can lead to accumulation in the gravity current nose of particles that originate well behind the gravity current head. However, efficient particle accumulation and transport with the speed c only occurs for particles with sufficient vertical buoyancy or vertical swimming capability. Weakly buoyant or neutral particles are swept down and back away from the gravity current head by the circulation in this region. A dam break into an ambient fluid with stratification (two‐layered) could result in the transport of some particles that are initially trapped in the forepart of the leading disturbance, an undular bore. However, this situation does not result in significant accumulation of particles at the leading edge of the disturbance. The numerical solutions show that particles that originate in the dense fluid ahead of the gravity current are much less likely to be accumulated at the gravity current nose. A simple scaling criterion for the minimum vertical buoyancy velocity for efficient accumulation and transport of buoyant particles based on the gravity current and particle characteristics is developed and compares favorably with the numerical results. The implications of these results for transport of larvae in coastal environments are discussed.

: 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.

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.

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 objective of this investigation is to examine the buoyancy-driven gravitational spreading currents, especially as applied to ocean disposal of wastewater and the accidental release of hazardous fluids, such as liquefied natural gas. A series of asymptotic solutions are used to describe the displacement of a gravitationally driven spreading front during an inertial phase of motion and the subsequent viscous phase. Solutions are derived by a force scale analysis and a self-similar technique for flows in stagnant, homogeneous, or linearly density-stratified environments. The self-similar solutions for inertial-buoyancy currents are found using an analogy to the well-known shallow-water wave propagation equations and also to those applicable to a blast wave in gasdynamics. For the viscous-buoyancy currents the analogy is to the viscous long wave approximation to a nonlinear diffusive wave, or thermal wave propagation. Other similarity solutions describing the initial stage of motion of the flow formed by the collapse of a finite volume fluid are developed by analogy to the expansion of a gas cloud into a vacuum. For the case of a continuous discharge there is initially a starting jet flow followed by the buoyancy-driven spreading flow. The jet mixing zone in such flows is described using Prandtl's mixing length theory. Dimensional analysis is used to derive the relevant scaling factors describing these flows.

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

Profiles of eddy viscosity and Prandtl mixing length in fluids flowing steadily and isothermally in smooth tubes have been calculated from the velocity data of several investigators for Reynolds numbers between 1.2 × 103 and 3.2 × 106. In the transition range unusually high values of eddy viscosity and mixing length are obtained in some portions of the stream. In the fully turbulent range the effect of Reynolds number is small and the mixing length tends toward zero at the center of the tube. The parameters for turbulent flow between parallel plates have been correlated through the concept of an equivalent tube. The results are of importance in designing equipment for heat and mass transfer and mixing.

Electromagnetic induction in the Vancouver Island region for a uniform inducing source field for 300 s period is investigated with the aid of three-dimensional (3-D) numerical and analogue model results and field site measurements. The thin sheet numerical model, based on the subducting Juan de Fuca plate analogue model ofDossoet al., consists of a 5km thick non-uniform thin sheet (comprising the lateral conductivity contrasts arising from the land, the varying depth ocean, and the sediment) underlain by a four-layer conductive structure. The four-layer conductive structure beneath the non-uniform thin sheet simulates the effect of the Juan de Fuca plate subducting Vancouver Island. To examine the effects of the ocean channel depth between Vancouver Island and the British Columbia (Canada) mainland, numerical results were obtained for two channel depths (0 and 600 m). The results indicate that the channel plays an important role in the geomagnetic response in the central and inner coastal regions of Vancouver Island. The general agreement of the 3-D numerical model induction arrows with the analogue model and field site induction arrows for 300 s supports the premise of a layered conductive substructure dipping at a small angle, at most, beneath Vancouver Island.

Self-similar gravity currents in porous media: Linear stability of the Barenblatt–Pattle solution revisited

The effect of the upstream conditions on propagation of gravity current over a slope is investigated using three-dimensional numerical simulations. The current produced by constant buoyancy flux, is simulated using a large eddy simulation solver. The dense saline solution used at the inlet is the driving force of the flow. Higher replenishment of the current is possible either by a high inflow discharge or high initial fractional density excess. In the simulations, it is observed that these two parameters affect the flow in different ways. Results show that the front speed of the descending current is proportional to the cube root of buoyancy flux, \((g_o^{\prime } Q)^{1/3}\), which agrees with the previous experimental and numerical observations. The height of the tail of the current grows linearly in the streamwise direction. Formation of a strong shear layer at the boundary of mixed upper layer and dense lower layer is observed within the body and the tail of the current. Over the tail of the current far enough from the inlet, the vertical velocity and density profiles are compared to the ones from an experimental study. Distance from the bed to the point of maximum velocity increases with an increase in inflow discharge, while it remains practically unchanged with increasing initial fractional excess density in the simulations. Even though the velocity profiles are in good agreement, some discrepancies are observed in fractional excess density profiles among experimental and numerical results. Possible reasons for these discrepancies are discussed. Generally, gravity current type of flows could be expressed in layer-integrated formulation of governing equations. However, layer integration introduces several constants, commonly known as shape factors, to the equations of motion. The values of these shape factors are calculated based on simulation results and compared to the values from experiments and to the favorably used ‘top hat’ assumption.

In this paper, a 3D forward curved (FC) centrifugal fan is simulated numerically to predict the turbulent flow and pressure field of the fan. Through this numerical simulation, the various turbulent models are compared and the influence of blade shapes on the fan performance is examined. For this means, four turbulence models (k-ε, k-ω, RSM, Spallart Allmaras) are applied to choose the most suitable one. Also for three kinds of blade shapes (flat blade, circular blade and NACA4412 airfoil blade), the flow fields are simulated separately to determine the most efficient blade shape. In numerical analysis, FVM (finite volume method) is used to solve the governing equations. Steady state conditions and multiple reference frames, including a MRF (moving reference frame) for the rotor region and a stationary reference frame for stator region, are used for numerical modeling. The numerical results are validated with experimental data. It is observed that k- ε is the most suitable turbulence model and the circular blade is the most efficient shape for this type of fan.