Momentumless Wake Research Articles

Two-Fluid Classical and Momentumless Laminar Far Wakes

Two-dimensional two-fluid classical and momentumless laminar far wakes are investigated in the boundary layer approximation. The velocity deficit satisfies a linear diffusion equation and the continuity equation in the upper and lower parts of the wakes. By using the multiplier method, conservation laws for the system of partial differential equations (PDEs) in the upper and lower parts of the wake are derived. Lie point symmetries associated with the conserved vectors for the classical and momentumless wakes are obtained. The conserved quantity for the two-fluid classical wake is the total drag on the obstacle, which is rederived. A new conserved quantity for the two-fluid momentumless wake is obtained, which satisfies the condition that the total drag on the obstacle is zero. Using the conserved quantities, it is shown that the equation of the interface is y=kx12, where k is a constant and x and y are Cartesian coordinates with origin at the trailing edge of the obstacle. New invariant solutions for the two-fluid classical and momentumless wakes with k=0 are found. Both solutions depend on the dimensionless parameter χ=(ρ1μ1)/(ρ2μ2) where suffices 1 and 2 refer to the upper and lower parts of the wake. For the special case in which the kinematic viscosity ratio ν2/ν1=1, two further solutions for the two-fluid momentumless wake are derived with k=±6.

Stability of momentumless wakes

The caudal fin of swimming animals can be modelled as a thrust-producing flapping foil. When considered alone, such a foil produces on average a jet wake with a positive net momentum. It has been argued that the instability characteristics of these averaged wakes are linked to the propulsion efficiency of swimming animals. Here, we reconsider this question by taking into account both the thrust and the drag exerted on a self-propelled swimming body. To do so, we study the stability of a family of momentumless wakes, constructed as the Oseen approximation of a force doublet moving at constant velocity. By performing a local stability analysis, we first show that these wakes undergo a transition from absolute to convective instability. Then, using the time stepper approach by integrating the linearised Navier–Stokes system, we investigate the global stability and reveal the influence of a non-parallel base flow as well as the role of the locally absolutely unstable upstream region in the wake. Finally, to complete the global scenario, we address the nonlinear evolution of the wake disturbance. These results are then discussed in the context of aquatic locomotion. According to the present stability results, and assuming the Oseen approximation whose validity has been assessed only for moderate Reynolds number, the momentumless wake of aquatic animals is generally stable, whereas the corresponding thrust part is unstable. It is therefore essential to consider all forces exerted on a self-propelled animal when discussing its wake stability and its propulsion efficiency.

Numerical analysis of turbulence decay in momentumless wakes behind a sphere and a prolate body of revolution

The numerical analysis of the characteristics of a turbulent flow in momentumless wakes behind a sphere and a prolate body of revolution in a homogeneous and linearly stratified media is performed. The results of the numerical simulation based on modern second- and third-order models of turbulence agree with the known theoretical and experimental data. The proximity between these flows in a number of inherent properties, including the self-similar turbulence decay in the far wake in a homogeneous medium and the interrelation between the second and third invariants of the Reynolds stress tensor are shown. The problem of the interaction between two regions of turbulent perturbations forming on the motion of a sphere and a prolate body of revolution in a homogeneous medium is considered.

Numerical simulation of momentumless turbulent wake dynamics in linearly stratified medium

This paper presents comparison of two numerical models of the momentumless turbulent wake dynamics behind a body of revolution in a linearly stratified medium, namely, the model based on direct (DNS) numerical integration of Navier–Stokes equations in the Oberbeck–Boussinesq approximation and the mathematical model with application of a semi-empirical turbulence model of the third order. The results of calculations by these two models agree with the known experimental data.

Dynamics of momentumless turbulent wake in a shear flow of a linearly stratified medium

The numerical model of momentumless turbulent wake in a horizontally homogeneous shear flow of linearly stratified medium has been constructed. Based on this model, the investigation of the wake dynamics has been performed. The obtained data demonstrate the transformation of the zone of turbulent perturbations and internal waves generated by the wake under the action of shear flow, which leads to the deceleration of turbulence decay at large time values after the body passage.

Dynamics of a momentumless turbulent wake in a shear flow

Numerical modeling and investigation of the dynamics of a momentumless turbulent wake in a horizontally homogeneous shear flow are performed. The obtained data demonstrate transformation of the turbulized region in awake under the action of a shear flow, and also substantial generation of turbulence energy by averaged motion, which leads to slower turbulence degeneracy at large time values after passage of the body. Some considerations on flow similarity by shear Froude number are presented along with calculation results illustrating this similarity.

The spatial evolution of fluctuations in a self-propelled wake compared to a patch of turbulence

The primary focus of this study is to contrast the influence of the mean velocity profile with that of the initial turbulence on the subsequent evolution of velocity and density fluctuations in a stratified wake. Direct numerical simulation is used to simulate the following cases: (a) a self-propelled momentumless turbulent wake, case SP50 with a canonical mean velocity profile, (b) a patch of turbulence, case TP1 with the same initial energy spectrum as (a), and (c) a patch of turbulence, case TP2 with a different initial energy spectrum with higher small-scale content. The evolution of the fluctuations is found to be strongly dependent on the initial energy spectrum, e.g., in case TP2, the kinetic energy is substantially smaller, and the late-wake vortices are less organized. The effect of the mean velocity field is negligible for mean kinetic energy (MKE) of the order 10% of the total kinetic energy and the evolution in this case is similar to a turbulent patch with the same initial energy spectrum. Increasing the MKE to 50% shows significant difference from the turbulent patch with the same initial energy spectrum during the initial stages of the evolution, but at later stages the evolution of turbulence statistics is similar. Both the turbulent patch and the momentumless wake show layering and formation of pancake eddies owing to buoyancy. Another objective of the paper is to compare the spatially evolving wake with the temporally evolving approximation when the initial near-wake condition of the temporal approximation is chosen to match the inflow of the spatially evolving model. The mean and turbulent flow statistics are found to agree well between the spatial and temporal computational models under these conditions.

Simulation of a propelled wake with moderate excess momentum in a stratified fluid

AbstractDirect numerical simulation is used to simulate the turbulent wake behind an accelerating axisymmetric self-propelled body in a stratified fluid. Acceleration is modelled by adding a velocity profile corresponding to net thrust to a self-propelled velocity profile resulting in a wake with excess momentum. The effect of a small to moderate amount of excess momentum on the initially momentumless self-propelled wake is investigated to evaluate if the addition of excess momentum leads to a large qualitative change in wake dynamics. Both the amount and shape of excess momentum are varied. Increasing the amount of excess momentum and/or decreasing the radial extent of excess momentum was found to increase the defect velocity, mean kinetic energy, shear in the velocity gradient and the wake width. The increased shear in the mean profile resulted in increased production of turbulent kinetic energy leading to an increase in turbulent kinetic energy and its dissipation. Slightly larger vorticity structures were observed in the late wake with excess momentum although the differences between vorticity structures in the self-propelled and 40 % excess momentum cases was significantly smaller than suggested by previous experiments. Buoyancy was found to preserve the doubly inflected velocity profile in the vertical direction, and similarity for the mean velocity and turbulent kinetic energy was found to occur in both horizontal and vertical directions. While quantitative differences were observed between cases with and without excess momentum, qualitatively similar evolution was found to occur.

Asymptotic and numerical analysis of momentumless turbulent wakes

The asymptotic behavior of turbulent axisymmetric and plane momentumless wakes was studied using the Reynolds-averaged momentum equations and the second-order model of turbulence. The similarity solutions were obtained analytically and the process of transition to self-similarity was studied numerically. It was found that a single-point spectrum of the solutions of the corresponding eigenvalue problem for turbulent energy and dissipation rate existed. However, the spectra of solutions for the normal components of the anisotropy tensor and for the mean velocity defect were discrete. The numerical solution of a non-self-similar problem shows, in accordance with experiments and analytic solutions, that mean and fluctuating velocities decay with different rates, shear stresses decay faster than normal stresses and the anisotropic component of normal stresses decays faster than the isotropic component. The analysis of solutions for a full system of the Reynolds stress equations showed the presence of ‘partial similarity’ of the turbulent momentumless wake when some variables (velocity defect and shear stress) remain non-similar in flow that is self-similar as a whole.

Numerical simulation of dynamics of turbulent wakes behind towed bodies in linearly stratified media

A hierarchy of semiempirical turbulence models of second order is involved for the description of a fluid flow in a far turbulent wake behind a towed body. The most complicated model includes the differential equations for normal Reynolds stresses transfer as well as the equation for the triple correlations of fluctuations of the vertical velocity component. Results of calculations are represented. They demonstrate the dynamics of a far turbulent wake in a linearly stratified medium in comparison with dynamics of a far momentumless turbulent wake. Anisotropic decay of turbulence in a far wake behind a towed body is numerically investigated. A numerical model of passive scalar dynamics in turbulent wakes behind bodies moving in a linearly stratified medium is represented.

Anisotropic turbulence decay in a far momentumless wake in a stratified medium

A new numerical model is constructed of turbulent wake dynamics behind bodies of revolution in a stably stratified medium based on a semiempirical third-order turbulence model. The employed differential transport equations of all triple correlations of velocity field oscillations, written with allowance for fourth-order cumulants and improved algebraic representations of joint triple correlations of velocity and density fluctuations, enable the comprehensive description of an anisotropic turbulence decay in a far momentumless wake in a stratified medium

Simulation of fish swimming and manoeuvring by an SVD-GFD method on a hybrid meshfree-Cartesian grid

In this paper we present the development of the Singular-Value Decomposition (SVD) based Generalized Finite Difference (GFD) method for the simulation of fluid–structure interaction (FSI) problems in a viscous fluid. The class of FSI problems is exemplified by the self-propulsion (swimming) and dynamic manoeuvring of deforming (undulating and flexing) bodies in a fluid medium. Computation is carried out on a hybrid grid comprising meshfree nodes around the undulating swimming body and Cartesian nodes in the background. The meshfree nodes are convected in tandem with the changing shape and motion of the swimming body. The resultant locomotion of the swimmer is governed by fully-coupled dynamic interaction between the deforming body and the fluid in accordance with Newton’s laws. Time integration of motion is carried out by a Crank–Nicolson based implicit iterative algorithm, which fully couples the changing position of the swimming body with the evolving flow field, for numerical stability. The numerical scheme is applied to the steady swimming/cruising and sharp turning manoeuvres of a two-dimensional carangiform fish. The Strouhal number approaches values for efficient steady swimming reported in Fish and Lauder (2006) and Triantafyllou and Triantafyllou (1993) [3,6] at high Reynolds number. An illustrative example shows the numerical carangiform swimmer executing a sharp turn through an angle of 70° from straight coasting within a space of about one body length. The results obtained are consistent with available literature. In steady swimming, the momentumless wake theoretically anticipated by Wu (2001) [57] is successfully reproduced here, as opposed to the inverse von Karman vortex street generally predicted by inviscid flow models. The momentumless wake, characterized by an aligned series of alternately-signed shed vortices, is symptomatic of a state of average equilibrium between drag acting on the body of the fish and thrust produced by its undulating tail fin. Guided swimming towards targets based on a simple feedback control scheme is also demonstrated.

Experimental investigation on the momentumless wake using trailing edge blowing

A developed plate stator model with and without trailing edge blowing (TEB) is studied using experimental methods. Wake characteristics of flow over the stator in the three-dimensional wake regimes are studied using hot-wire anemometry (HWA) and particle image velocimetry (PIV) techniques. First, the mean velocity profiles have been measured in the wake of the stator using HWA. Four wake characteristics have been obtained through momentum thickness judgments: pure wake, weak wake, momentumless wake, and jet. These velocity profiles show some differences in momentum deficit for the four cases. Then, the velocity spectra of the pure wake and momentumless wake obtained through the HWA measurements showed that TEB can eliminate the shedding vortex of the stator. Characteristic length scales based on the wake turbulent intensity profiles showed that the momentumless wake can reduce the wake width and depth. PIV measurement is carried out to measure the flow field of the four wakes. Finally, the application of TEB approaching momentumless wake status is used on an industrial ventilation low-pressure axial fan to assess noise reduction. The results show that TEB can make the outlet of the stator uniform, reduce velocity fluctuation, destroy the vorticity structure downstream of the stator, and reduce interaction noise level of the stator and rotor.

Dynamics of turbulent wake with small excess momentum in stratified media

A numerical model of turbulent wake dynamics with small excess momentum in linearly stratified fluid has been created on the basis of a three-dimensional parabolized system of averaged equations for motion, continuity, incompressibility, energy of turbulence and the rate of dissipation transfer. The calculation results show that as in the case of homogeneous fluid, the wake’s excess momentum of the order of ±5% and ±10% off the total momentum in the drag wake has a weak influence on the decay of the turbulent energy and more significant influence on the axis value of the defect of the longitudinal velocity component. Turbulent wakes with small excess momentum generate internal waves which differ only insignificantly from the waves generated by momentumless wakes.

Numerical models of the second and third orders for a momentumless turbulent wake dynamics in a linearly stratified medium

Numerical analysis of the evolution of a momentumless turbulent wake in a linearly stratified medium has been performed using mathematical models based on improved representations of ternary correlations of turbulent pulsing of the velocity and density fields. The results of calculations are compared to experimental data.

On wake response to asymmetric blowing in spurts from a blunt trailing edge

In the present investigation, a technique has experimentally been evolved with the aim to control the unsteady wake dominated by periodic vortex-shedding. The technique uses pulsating surface blowing that is applied asymmetrically just before separation from the blunt trailing edge of a thick aerofoil model. Thus, momentum is injected in spurts in only one of the two separated shear layers. The control parameters are the mass flow rate and the forcing frequency of the pulsating blowing. The technique is particularly effective in suppressing the vortex-shedding when the blowing is moderate and the forcing frequency is twice the natural vortex-shedding frequency. As a consequence of injecting momentum in spurts, there is a significant saving in mass flow required to achieve a condition of the momentumless wake. In spite of the initial conditions being strongly asymmetric, the flow pattern in the wake was observed to quickly assume a remarkable symmetry.

Numerical modelling of momentumless wakes using semiempirical turbulence models of second and third order

In this paper we present numerical models of dynamics of a momentumless turbulence wake behind a body of revolution, which are based on improved second- and third-order turbulence models. We compare data of numerical experiments with experimental data.

Compatible differential constraints to an infinite chain of transport equations for cumulants

We study an infinite chain of transport equations for cumulants which appears in modeling the dynamics of a momentumless turbulent planar wake. The method of compatible differential constraints for formulating its integrability properties is applied. The compatibility conditions obtained make it possible to realize a reduction of the original infinite chain of transport equations and to present an algorithm for calculating cumulants of arbitrary order.

Passive scalar dynamics in turbulent wakes of bodies moving in a linearly stratified medium

A numerical model of passive scalar dynamics in turbulent wakes of bodies of revolution in a linearly stratified medium is constructed. To describe the flow we use a three-dimensional parabolized system of equations of motion, continuity, incompressibility, turbulent energy transport, dissipation rate, averaged concentration and dispersion of passive scalar concentration fluctuations. Computational results demonstrate the behaviour of passive scalar characteristics both for momentumless wake and in the wake behind a towed body.

Two-fluid jets and wakes

Similarity solutions for laminar two-fluid jets and wakes are derived in the boundary-layer approximation. Planar and axisymmetric fan jets as well as classical and momentumless planar wakes are considered. The interface between the immiscible fluids is stabilized by the action of gravity, with the heavier fluid, taken to be a liquid, placed beneath the lighter fluid. Velocity profiles for the jets and the classical wake depend intimately, but differently, on the parameter χ=ρ1μ1/ρ2μ2, where ρi and μi are, respectively, the density and absolute viscosity of the fluid in the upper (i=1) and lower (i=2) fluid domains, while the momentumless wake profile depends on the parameter Ω=ρ1μ23/ρ2μ13. Generally, all interfaces deflect from horizontal except the fan jet. However, while the interface for the classical planar two-fluid wake is never flat, the interfaces for the planar jet and the momentumless wake become flat in the particular case μ1=μ2. Velocity profiles illustrating the strongly asymmetrical jet and wake profiles that arise in air-over-water, oil-over-water, and air-over-oil flows are presented.