Shear Of Basic Flow Research Articles

Baroclinic Instability of a Time-Dependent Zonal Shear Flow

In the real atmosphere, the development of large-scale motion is often related to the baroclinic properties of the atmosphere. So, it is necessary to discuss the stability condition of baroclinic flow. It is advantageous to use a layered model to discuss baroclinic instability, not only to apply the potential vortex equation directly, but also to deal with shear of basic flow. The stability and oscillatory shear ability of Rossby waves are studied based on the two-layer Phillips model in the β plane; then, we summarize the baroclinic instability of time-dependent zonal shear flows. The multiscale method is used to eliminate some terms of natural frequency oscillations of nonlinear operators in the third-order expansion, thus generating an equation about the amplitude of the lowest-order Rossby wave in the long-time variable. The large amplitude perturbation begins to decrease, which produces the desired behavior. After the amplitude decreases for some time, the amplitude of Rossby waves can still be found to oscillate periodically with the time variable.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Coherent structures of nonlinear barotropic-baroclinic interaction in unequal depth two-layer model

The nonlinear barotropic-baroclinic interaction theory is of great importance for the understanding of the atmosphere and the oceans. In this paper, a mathematical model based on analytical methods is established to discuss the propagation of the coherent structure of the nonlinear barotropic-baroclinic interaction of two-layer fluids in geophysics. It is the first time that the coupled Boussinesq equations are derived by using perturbation and spatiotemporal extensions transformations to simulate the evolutions of nonlinear barotropic and baroclinic waves. The solitary wave solutions are analytically obtained to explain the excitation, propagation and decrease of the coherent structures. The physical mechanism of the nonlinear wave-wave coherent structures is discussed by considering the effects of different physical factors, such as the topography, the Earth’s rotation and the basic shear flow. It is found that both topography and beta parameter have essential effects on the evolutions of the coherent structures. Furthermore, the effect of the unequal-depth parameter on the propagation of coherent structures is discussed.

Optimal perturbations in viscous channel flow with crossflow

This work is devoted to the studies of optimal perturbation and its transient growth characteristics in the Poiseuille flow with Reynolds number R = 1000 and with crossflow. The crossflow Reynolds number R v , based on the uniform velocity in the negative normal direction, is varied from 0 to 200. The numerical result shows that the crossflow enhances the transient growth for three-dimensional disturbance case when . Seen from the contours of optimal energy growth in the wave number plane, the optimal perturbation that corresponds to the peak value of optimal growth becomes oblique perturbation, when crossflow is introduced. The corresponding streamwise wave number α P increases continuously with increasing R v , while spanwise wave number β P drops firstly with increasing R v and later increases. The comparison of exponential growth and transient growth is performed. It is observed that exponential growth becomes profound and even predominant over transient growth after a short period when the crossflow is sufficiently strong. Moreover, the mechanism of transient growth is discussed and it is found that the uniform crossflow can not cause transient growth in the absence of shear in basic flow. With increasing R v , the Orr mechanism becomes more and more important relative to lift-up mechanism for the transient growth of optimal perturbation.

Second-Order Phase Transition in Counter-Rotating Taylor-Couette Flow Experiment.

In many basic shear flows, such as pipe, Couette, and channel flow, turbulence does not arise from an instability of the laminar state, and both dynamical states co-exist. With decreasing flow speed (i.e., decreasing Reynolds number) the fraction of fluid in laminar motion increases while turbulence recedes and eventually the entire flow relaminarizes. The first step towards understanding the nature of this transition is to determine if the phase change is of either first or second order. In the former case, the turbulent fraction would drop discontinuously to zero as the Reynolds number decreases while in the latter the process would be continuous. For Couette flow, the flow between two parallel plates, earlier studies suggest a discontinuous scenario. In the present study we realize a Couette flow between two concentric cylinders which allows studies to be carried out in large aspect ratios and for extensive observation times. The presented measurements show that the transition in this circular Couette geometry is continuous suggesting that former studies were limited by finite size effects. A further characterization of this transition, in particular its relation to the directed percolation universality class, requires even larger system sizes than presently available.

On the spectral instability and bifurcation of the 2D-quasi-geostrophic potential vorticity equation with a generalized Kolmogorov forcing

In this article, the spectral instability and the associated bifurcations of the shear flows of the 2D quasi-geostrophic equation with a generalized Kolmogorov forcing are investigated. To determine the linear instability of the basic shear flow, we write the corresponding eigenvalue problem as a system of finite difference equations whose nontrivial solutions are expressed in the form of continued fractions. By a rigorous analysis of these continued fractions, we prove the existence of a number Rc such that if the control parameter R, which is proportional to the Reynolds number and the intensity of the curl of forcing, is over Rc, then the basic shear flow loses its stability. To shed light on the bifurcation involved in the loss of stability of the basic shear flow, a natural method is used to reduce the quasi-geostrophic equation to a system of ordinary differential equations. Based on numerical experiments on the coefficients of this reduced system, we show that both supercritical and subcritical Hopf bifurcations occur depending on the frequency of the generalized Kolmogorov forcing. Moreover, we investigate the double Hopf bifurcations which occur at critical aspect ratios. Our results show that in the double Hopf bifurcation case, two periodic solutions, one stable and the other unstable, bifurcate on R>Rc.

Instability mechanisms for thermocapillary flow in an annular pool heated from inner wall

The linear stability analysis of thermocapillary flow in an annular pool heated from inner wall was performed by using spectral element method. The physical instability mechanisms for different Prandtl numbers (Pr) ranging from 0.001 to 1.4 were studied by energy analysis under critical mode. We observed three instability types: (i) the Hopf (oscillatory) bifurcation with wavenumber k = 18 for Pr < 0.02, and the corresponding neutral mode is amplified due to the basic shear flow; (ii) the stationary instability with k = 3 or k = 2 for 0.02 < Pr < 0.6; (iii) the Hopf bifurcation again with k = 3 for 0.6 < Pr < 1.4. In both Prandtl number ranges of 0.02 < Pr < 0.6 and 0.6 < Pr < 1.4, thermocapillary force induced by the disturbance temperature field drives primarily the flow instability.

Dynamic Transitions and Baroclinic Instability for 3D Continuously Stratified Boussinesq Flows

The main objective of this article is to study the nonlinear stability and dynamic transitions of the basic (zonal) shear flows for the three-dimensional continuously stratified rotating Boussinesq model. The model equations are fundamental equations in geophysical fluid dynamics, and dynamics associated with their basic zonal shear flows play a crucial role in understanding many important geophysical fluid dynamical processes, such as the meridional overturning oceanic circulation and the geophysical baroclinic instability. In this paper, first we derive a threshold for the energy stability of the basic shear flow, and obtain a criteria for nonlinear stability in terms of the critical horizontal wavenumbers and the system parameters such as the Froude number, the Rossby number, the Prandtl number and the strength of the shear flow. Next we demonstrate that the system always undergoes a dynamic transition from the basic shear flow to either a spatiotemporal oscillatory pattern or circle of steady states, as the shear strength $\Lambda$ of the basic flow crosses a critical threshold $\Lambda_c$. Also we show that the dynamic transition can be either continuous or catastrophic, and is dictated by the sign of a transition parameter $A$, fully characterizing the nonlinear interactions of different modes. A systematic numerical method is carried out to explore transition in different flow parameter regimes. We find that the system admits only critical eigenmodes with horizontal wave indices $(0,m_y)$. Such modes, horizontally have the pattern consisting of $m_y$-rolls aligned with the x-axis. Furthermore, numerically we encountered continuous transitions to multiple steady states, continuous and catastrophic transitions to spatiotemporal oscillations.

Rheological Properties of Collagen Matter Predicted Using an Extrusion Rheometer

AbstractThis article reports on measurements of the rheological properties of slightly compressible collagen liquid (from 6.6 to 8.0% mass fraction of native bovine collagen in water) using a capillary rheometer. The extrusion rheometer is equipped with an annular slit between a circular tube and a central pin. The sample of collagen to be tested is pushed by a piston from a container to the annular measuring section. A manually adjusted hydraulic drive enables the piston velocity and the corresponding shear rate to be varied continuously within the range from 50 to 3000 s−1. Strain gages installed in the outer cylinder monitor the axial pressure profile. It is therefore possible to identify not only the basic nonelastic shear flow characteristics (Herschel Bulkley and Power law model parameters), but also the first normal difference evaluated from the exit pressure. The results of experiments carried out at room temperature show rather broad variability of the parameters.Practical ApplicationsThe rheological properties of native collagenous materials are of primary importance in the design of extrusion processes, for example, extrusion of sausage casings (a description of the flow behavior not only inside the screw extruders, but also while the extruded tubes are being blown) and similar technologies (injection molding, film blowing, rolling, and so forth). Knowledge of the rheological properties of native collagenous vascular grafts is important for an indirect assessment of irradiation effects and the effects of additives on a collagenous matrix (crosslinking). Native collagenous materials are rather “awkward” from the point of view of rheometry. For example, rotational rheometers are limited to a linear oscillation regime (due to discharge of the samples and shear banding), and the application of capillary rheometers is complicated by the inhomogeneity of native collagen (presence of bubbles). The proposed rheometer takes into account the compressibility of the tested samples, and enables an evaluation not only of the shear properties but also of the elastic properties within the range of deformation rates corresponding to the production lines.

Directed percolation phase transition to sustained turbulence in Couette flow

Decades-old speculation that the transition to turbulence belongs to the directed percolation universality class is confirmed with experimental and numerical data for flow through a quasi-one-dimensional Couette geometry. Turbulence is one of the most frequently encountered non-equilibrium phenomena in nature, yet characterizing the transition that gives rise to turbulence in basic shear flows has remained an elusive task. Although, in recent studies, critical points marking the onset of sustained turbulence1,2,3 have been determined for several such flows, the physical nature of the transition could not be fully explained. In extensive experimental and computational studies we show for the example of Couette flow that the onset of turbulence is a second-order phase transition and falls into the directed percolation universality class. Consequently, the complex laminar–turbulent patterns distinctive for the onset of turbulence in shear flows4,5 result from short-range interactions of turbulent domains and are characterized by universal critical exponents. More generally, our study demonstrates that even high-dimensional systems far from equilibrium such as turbulence exhibit universality at onset and that here the collective dynamics obeys simple rules.

Dissipative Nonlinear Schrödinger Equation with External Forcing in Rotational Stratified Fluids and Its Solution**Supported by Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant No. 41421005, National Natural Science Foundation of China under Grant Nos. 41376030, 41376029, 41476019, NSFC–Shandong Joint Fund for Marine Science Research Centers Grant (U1406401), Special Funds for Theoretical Physics of

The dissipative nonlinear Schrödinger equation with a forcing item is derived by using of multiple scales analysis and perturbation method as a mathematical model of describing envelope solitary Rossby waves with dissipation effect and external forcing in rotational stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt–Vaisala frequency and β effect are important factors in forming the envelope solitary Rossby waves. By employing Jacobi elliptic function expansion method and Hirota's direct method, the analytic solutions of dissipative nonlinear Schrödinger equation and forced nonlinear Schrödinger equation are derived, respectively. With the help of these solutions, the effects of dissipation and external forcing on the evolution of envelope solitary Rossby wave are also discussed in detail. The results show that dissipation causes slowly decrease of amplitude of envelope solitary Rossby waves and slowly increase of width, while it has no effect on the propagation speed and different types of external forcing can excite the same envelope solitary Rossby waves. It is notable that dissipation and different types of external forcing have certain influence on the carrier frequency of envelope solitary Rossby waves.

Nonlinear shearing modes approach to the diocotron instability of a planar electron strip

The nonlinear evolution of the diocotron instability of a planar electron strip is investigated analytically by means of the nonlinear shearing mode for the solution of the initial and boundary value problems. The method is based on the sheared spatial coordinates which account for the motion of electron flow in the electrostatic field of the unstable diocotron modes in addition to the unperturbed sheared motion of the electron flow on the transformed shear coordinates. The time evolutions are studied by the solution of the initial and boundary value problems. The obtained solutions for the perturbed electrostatic potential include two nonlinear effects—the effect of the distortion of the boundaries of the planar electron strip and the effect of the coupling of the sheared nonmodal diocotron modes. It was proved by a two-dimensional particle-in-cell simulation that the developed theory is valid as long as the distortion of the boundaries of the basic shear flow does not change the frequency and growth rate of the linear diocotron instability in the transformed coordinates.

Dissipative Nonlinear Schrödinger Equation for Envelope Solitary Rossby Waves with Dissipation Effect in Stratified Fluids and Its Solution

We solve the so-called dissipative nonlinear Schrödinger equation by means of multiple scales analysis and perturbation method to describe envelope solitary Rossby waves with dissipation effect in stratified fluids. By analyzing the evolution of amplitude of envelope solitary Rossby waves, it is found that the shear of basic flow, Brunt-Vaisala frequency, andβeffect are important factors to form the envelope solitary Rossby waves. By employing trial function method, the asymptotic solution of dissipative nonlinear Schrödinger equation is derived. Based on the solution, the effect of dissipation on the evolution of envelope solitary Rossby wave is also discussed. The results show that the dissipation causes a slow decrease of amplitude of envelope solitary Rossby waves and a slow increase of width, while it has no effect on the propagation velocity. That is quite different from the KdV-type solitary waves. It is notable that dissipation has certain influence on the carrier frequency.

Influence of vertical shear of basic tangential wind on the development and maintenance of typhoon

By using a linear symmetric Conditional Instability of Second Kind (CISK) model containing basic flow, we study the interactions between basic flow and mesoscale disturbances in typhoon. The result shows that in the early stage of typhoon formation, the combined action of vertical shear of basic flow at low level and CISK impels the disturbances to grow rapidly and to move toward the center of typhoon. The development of disturbances, likewise, influences on typhoon’s development and structure. Analysis of the mesoscale disturbances’ development and propagation indicates that the maximum wind region moves toward the center, wind velocity increases, and circulation features of an eye appear. Similarly, when a typhoon decays, the increase of low-level vertical wind shear facilitates the development of mesoscale disturbances. In turn, these mesoscale disturbances will provide typhoon with energy and make the typhoon intensify again. Therefore, it can be said that typhoon has the renewable or self-repair function.

The disturbance energy of Rossby wave in barotropic atmosphere

Fourier-transformed method is used to investigate the problem of the rapid growth and decay of the disturbance energy of Rossby wave in barotropic atmosphere on the assumption that this kind of atmosphere is adiabatic, non-frictional, unforced, non-dissipative and quasigeostrophic on a beta plane. The analytical solution of linear barotropic potential vorticity equation is obtained for the first time and we go a step further to discuss the relations between the disturbance energy and the initial perturbation flow wave numbers, shear of basic flow and viscid coefficient by using the above analytical solution.

Transient growth in a two-fluid channel flow under normal electric field

A linear model of a layered channel flow of two perfectly dielectric viscous fluids in the presence of uniform normal electric field is built. The effect of the normal electric field on transient growth of small disturbances is studied at two values of Weber number. The numerical result shows that the electric field enhances the transient growth for both two-dimensional and three-dimensional disturbance cases. The contours of optimal energy growth are represented in the wave number plane. When the electric field is small, the optimal disturbance that corresponds to the peak value of optimal growth is two dimensional. It is governed by the lift-up mechanism and is little influenced by the electric field. However, when the electrical Euler number exceeds a critical value, the optimal disturbance is three dimensional with streamwise uniform wave number and is partially dominated by the electric field, and moreover, the spanwise wave number has a linear relationship with the electrical Euler number. The comparison of exponential growth and transient growth is performed. It is shown that exponential growth becomes profound and even predominant over transient growth when the electric field is sufficiently strong. In addition, the mechanism of transient growth is discussed and it is found that the existence of material interface may cause transient growth in the absence of shear in basic flow.

MKdV Equation for the Amplitude of Solitary Rossby Waves in Stratified Shear Flows with a Zonal Shear Flow

AbstractIn this paper, modified Korteweg-de Vries (mKdV) equations for the amplitude of solitary Rossby waves in stratified fluids with a zonal shear flow are derived by using a weakly nonlinear method. It is found that the coefficients of mKdV equations depend not only on the β effect and the Vaisala-Brunt frequency, but also on the basic shear flow.

Stability of an erodible bed in various shear flows

The 2D laminar quasi-steady asymptotically simplified and linearized flow with a simplified mass transport of sediments is solved over a slowly erodible bed in various laminar basic shear flow (steady, oscillating or decelerating). The simplified mass transport equation includes the two following phenomena: flux of erosion when the skin friction goes over a threshold value, and a non local effect coming either from an inertial effect or from a slope effect. It is shown that the bed is always unstable for small wave numbers. Examples of long time evolution in various shear regimes are presented, wave trains of ripples are created and merge into a unique bump. This coarsening process is such that the maximum wave length obeys a power law with time.

Boundary-layer separation and adverse pressure gradient for 2-D viscous incompressible flow

We study the detailed process of bifurcation in the flow’s topological structure for a two-dimensional (2-D) incompressible flow subject to no-slip boundary conditions and its connection with boundary-layer separation. The boundary-layer separation theory of M. Ghil, T. Ma and S. Wang, based on the structural-bifurcation concept, is translated into vorticity form. The vorticity formulation of the theory shows that structural bifurcation occurs whenever a degenerate singular point for the vorticity appears on the boundary; this singular point is characterized by nonzero tangential second-order derivative and nonzero time derivative of the vorticity. Furthermore, we prove the presence of an adverse pressure gradient at the critical point, due to reversal in the direction of the pressure force with respect to the basic shear flow at this point. A numerical example of 2-D driven-cavity flow, governed by the Navier Stokes equations, is presented; boundary-layer separation occurs, the bifurcation criterion is satisfied, and an adverse pressure gradient is shown to be present.

On Lagrangian time scales and particle dispersion modeling in equilibrium turbulent shear flows

As intermediate quantities available from various existing numerical computations, the fluid Lagrangian time scales are of primary importance in the development of probability density function models for turbulent flows. Similarly, the time scales of the fluid seen by discrete particles in two-phase flows are essential for the development of dispersion models based on stochastic differential equations. Such time scales are obviously depending not only on the particle properties but also on the fluid Lagrangian and Eulerian time scales. A model is proposed here to estimate the directional dependence of the fluid Lagrangian time scales and drift coefficients in one-directional equilibrium turbulent shear flows, based on a local homogeneity assumption in the frame of the generalized Langevin model. Through comparison with available direct and large eddy simulation predictions in channel flows and in a homogeneous shear flow, the model is shown to lead to significant improvements in the streamwise and spanwise directions, where the existing empirical laws for the Lagrangian time scales are far from being satisfactory. We examine the way this model can be used to build a suitable stochastic process for the fluid seen by inertial particles in such basic turbulent shear flows.