Non-rotating Flow Research Articles

Orographically generated nonlinear waves in rotating and non-rotating two-layer flow

This paper reports experimental observations of finite amplitude interfacial waves forced by a surface-mounted obstacle towed through a two-layer fluid both when the fluid is otherwise at rest and when the fluid is otherwise rotating as a solid body. The experimental apparatus is sufficiently wide so that sidewall effects are negligible even in near-critical flow when the towing speed is close to the interfacial long-wave speed and the transverse extent of the forced wavefield is large. The observations are modelled by a simple forced Benjamin–Davis–Acrivos equation and comparison between integrations of both linear and nonlinear problems shows the fundamental nonlinearity of the near-critical flow patterns. In both the experiments and integrations rotation strongly confines the wavefield to extend laterally over distances only of order of the Rossby radius and also introduces finite-amplitude sharply pointed lee waves in supercritical flow.

Rotating gravity currents. Part 1. Energy loss theory

A comprehensive energy loss theory for gravity currents in rotating rectangular channels is presented. The model is an extension of the non-rotating energy loss theory of Benjamin (J. Fluid Mech. vol. 31, 1968, p. 209) and the steady-state dissipationless theory of rotating gravity currents of Hacker (PhD thesis, 1996). The theory assumes the fluid is inviscid, there is no shear within the current, and the Boussinesq approximation is made. Dissipation is introduced using a simple method. A head loss term is introduced into the Bernoulli equation and it is assumed that the energy loss is uniform across the stream. Conservation of momentum, volume flux and potential vorticity between upstream and downstream locations is then considered. By allowing for energy dissipation, results are obtained for channels of arbitrary depth and width (relative to the current). The results match those from earlier workers in the two limits of (i) zero rotation (but including dissipation) and (ii) zero dissipation (but including rotation). Three types of flow are identified as the effect of rotation increases, characterized in terms of the location of the outcropping interface between the gravity current and the ambient fluid on the channel boundaries. The parameters for transitions between these cases are quantified, as is the detailed behaviour of the flow in all cases. In particular, the speed of the current can be predicted for any given channel depth and width. As the channel depth increases, the predicted Froude number tends to , as for non-rotating flows.

Laboratory Studies of Low Vortices in a Low Froude Number FlowPasting over an Isolated Obstacle

The present paper mainly focuses on the generation and developmentof lee vortices for the low Froude number density stratified flowpassing over an isolated obstacle. A series of laboratory experimentswere carried out by towing-tank equipments to simulate the airflowpassing over a large terrain. Based on the similarity principle, theFroude number of the stratified flow, the stratified condition, and therotation parameter are chosen in the specified scope. Our studies showthat the lee vortices can appear in the non-rotational and rotationalcases. For the non-rotational fluid, the lee vortices generate onlyunder the condition of strong stratification and the small Froudenumber (0.1<Fr<0.3). However, the lee vortices can be formed underthe condition of relatively strong stratification and the properFroude number (0.1<Fr<0.5) for the rotational fluid. Theresults of the divided experiments including non-rotational androtational cases show that the formation mechanism of lee vortices isdifferent for non-rotational flow and rotational flow. The leevortices are created by the tilting term of perturbation velocity forthe non-rotational flow, but for the rotational flow, the β-effectassociated with the horizontal convergence and divergence of thehorizontal perturbation velocity is more important.

A friction factor bound for transitional pipe flow

An upper bound λmax on the friction factor λ is found for transitional pipe flow up to Reynolds numbers of Re=4000. This bound corresponds to the laminar Hagen–Poiseuille value until the energy stability point of Re=81.52 (where λmax=0.7851) after which it decreases monotonically to approach a viscosity-independent asymptote of 0.27±0.01 as Re→∞. Comparison is made with the friction factors associated with recently discovered, finite-amplitude traveling waves in rotating and nonrotating pipe flow as well as experimental data.

A model for quasi-spherical magnetized accretion flow

A model for axisymmetric magnetized accretion flow is proposed. The dominant mechanism of energy dissipation is assumed to be the magnetic diffusivity due to turbulence in the accretion flow. In analogy to the advection-dominated accretion flow (ADAF) solutions, a constant fraction of the resistively dissipated energy is stored in the accreting gas and the rest is radi- ated. We first introduce the general self-similar solutions which describe a resistive and nonrotating flow with purely poloidal magnetic field. The radial dependence of physical quantities is identical to that in viscous ADAF solutions. Although the main focus of this study is on nonrotating magnetized accretion flow, for rotating flow with both poloidal and toroidal components of magnetic field we find a radial scaling of solutions similar to the nonrotating case. We show that the accretion and the rotation velocities are both below the Keplerian rate, irrespective of the amount of cooling. We show that the set of equations is reduced to one second order differential equation for a nonrotating flow. The geometrical shape of the disk changes depending on the fraction of the resistively dissipated energy which is stored in the accreting gas. However, there is a hot low-density gas above the disk in almost all cases. The net accretion rate is calculated for a set of illustrative parameters.

Inertial oscillations and frontogenesis driven by a quadratic horizontal density variation

We consider inviscid rotating flow driven by a horizontally quadratic density variation in a horizontally unbounded slab. This configuration permits a similarity solution, removing the dependence on the horizontal coordinate from the vorticity and temperature equations, which are then solved by numerical integration along characteristics. At large values of Rossby number, the flow proceeds to a singularity in a similar manner to the non-rotating flow with the same initial conditions. At small values of Rossby number there are inertial oscillations of growing amplitude, which have been analysed using the method of multiple scales. The oscillations become desynchronised between the upper and lower parts of the domain, and static instability appears for a small fraction of each oscillation period. Eventually the oscillations give way to the rapid formation of a singularity, in contrast to geostrophic adjustment theory which predicts that a singularity will form only if the Rossby number is sufficiently large.

Computational assessment of subcritical and delayed onset in spiral Poiseuille flow experiments

For spiral Poiseuille flow with radius ratios $\eta \equiv R_i/R_o = 0.77$ and 0.95, we have computed complete linear stability boundaries, where $R_i$ and $R_o$ are the inner and outer cylinder radii, respectively. The analysis accounts for arbitrary disturbances of infinitesimal amplitude over the entire range of Reynolds numbers $Re$ for which the flow is stable for some range of Taylor number $Ta$, and extends previous work to several non-zero rotation rate ratios $\mu \equiv \Omega_o/\Omega_i$, where $\Omega_i$ and $\Omega_o$ are the (signed) angular speeds. For each combination of $\mu$ and $\eta$, there is a wide range of $Re$ for which the critical $Ta$ is nearly independent of $Re$, followed by a precipitous drop to $Ta = 0$ at the $Re$ at which non-rotating annular Poiseuille flow becomes unstable with respect to a Tollmien–Schlichting-like disturbance. Comparison is also made to a wealth of experimental data for the onset of instability. For $Re > 0$, we compute critical values of $Ta$ for most of the $\mu = 0$ data, and for all of the non-zero-$\mu$ data. For $\mu = 0$ and $\eta = 0.955$, agreement with data from an annulus with aspect ratio (length divided by gap) greater than 570 is within 3.2% for $Re \leq 325$ (based on the gap and mean axial speed), strongly suggesting that no finite-amplitude instability occurs over this range of $Re$. At higher $Re$, onset is delayed, with experimental values of $Ta_{\hbox{\scriptsize{\it crit}}}$ exceeding computed values. For $\mu = 0$ and smaller $\eta$, comparison to experiment (with smaller aspect ratios) at low $Re$ is slightly less good. For $\eta = 0.77$ and a range of $\mu$, agreement with experiment is very good for $Re < 135$ except at the most positive or negative $\mu$ (where $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$), whereas for $Re \geq 166$, $Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it expt}}} > Ta_{\hbox{\scriptsize{\it crit}}}^{\hbox{\scriptsize{\it comp}}}$ for all but the most positive $\mu$. For $\eta = 0.9497$ and 0.959 and all but the most extreme values of $\mu$, agreement is excellent (generally within 2%) up to the largest $Re$ considered experimentally (200), again suggesting that finite-amplitude instability is unimportant.

Topographic and boundary effects on steady and unsteady flow through straits

An overview of topographic and boundary effects in flows through straits is presented. The emphasis is on the various types of interaction of uniform and stratified flow with the topography of straits, especially the interaction with sills. The interaction of a single-layer flow in a channel with a sill (obstacle) depends on the Froude number of the flow and the relative height of the sill. Two classification schemes (one steady and one pseudo-steady) are given. For single-layer flow in straits where the effects of rotation are important a further parameter, the ratio of the channel width to a Rossby radius, is needed and a similar classification scheme as for the non-rotating flows can be obtained. Unsteady flows often result in bores (for single-layer flows) and internal bores (in stratified flows) and the behaviour of these is discussed. The flow of stratified flow past sills may generate lee waves, while the increased flow speeds at the sill may allow deep water to be drawn up and over the sill (aspiration). Features of the strait topography away from the sill, including slopes at the side of the channel and also downstream of the sill, channel curvature and channel widening have important effects on the character of the flow and on mixing.

DNS of Turbulent Ekman Boundary Layer over a Wavy Terrain.

The direct numerical simulation (DNS) is performed on the turbulent Ekman boundary layer over wavy terrain. Hills are arranged perpendicularly to the direction of the geostrophic wind. Turbulent statistics such as the mean velocity profile and the velocity fluctuations are obtained. The thickness of the Ekman boundary layer is significantly increased compared with the flow on the flat ground. The instantaneous turbulent structures are visualized using high and low speed streaks. The streaks become segmented in the cases of the wavy terrain. The Reynolds stress, and the turbulent kinetic energy production, and their production rate are obtained and discussed in comparison with those of non-rotating flow.

Electromotive force and large-scale magnetic dynamo in a turbulent flow with a mean shear.

An effect of sheared large-scale motions on a mean electromotive force in a nonrotating turbulent flow of a conducting fluid is studied. It is demonstrated that in a homogeneous divergence-free turbulent flow the alpha effect does not exist, however a mean magnetic field can be generated even in a nonrotating turbulence with an imposed mean velocity shear due to a "shear-current" effect. A mean velocity shear results in an anisotropy of turbulent magnetic diffusion. A contribution to the electromotive force related to the symmetric parts of the gradient tensor of the mean magnetic field (the kappa effect) is found in nonrotating turbulent flows with a mean shear. The kappa effect and turbulent magnetic diffusion reduce the growth rate of the mean magnetic field. It is shown that a mean magnetic field can be generated when the exponent of the energy spectrum of the background turbulence (without the mean velocity shear) is less than 2. The shear-current effect was studied using two different methods: the tau approximation (the Orszag third-order closure procedure) and the stochastic calculus (the path integral representation of the solution of the induction equation, Feynman-Kac formula, and Cameron-Martin-Girsanov theorem). Astrophysical applications of the obtained results are discussed.

Scaling in three-dimensional and quasi-two-dimensional rotating turbulent flows

We have made velocity time series measurements (using hot film probes) and velocity field measurements (using particle image velocimetry) on turbulent flow in a rotating annulus. For low annulus rotation rates the Rossby number was of order unity and the flow was three-dimensional (3D), but at high rotation rates the Rossby number was only about 0.1, comparable to the value for oceans and the atmosphere on large length scales. The low Rossby number (quasi-geostrophic) flow was nearly two-dimensional (2D), as expected from the Taylor–Proudman theorem. For the 3D flow we found that the probability distribution function (PDF) for velocity differences along the direction of the flow, δv(d)=v(x0+d)−v(x0), was Gaussian for large separations d and non-Gaussian (with exponential tails) for small d, as has been found for nonrotating turbulent flows. However, for low Rossby number flow, the PDF was self-similar (independent of d) and non-Gaussian. The exponents characterizing the structure functions, Sp=〈(δv)p〉∼dζp were obtained by the extended self-similarity method. For 3D flow the exponents departed from p/3 with increasing p, as has been found for turbulence in nonrotating flows, while for the quasi-2D turbulent flow, the exponents increased linearly with p, as expected for a self-similar flow. We applied the β-test of the hierarchical structure model [She and Lévêque, Phys. Rev. Lett. 72, 336 (1994)] and found that β remained constant at β≃0.75 as the rotation was increased from the 3D to the 2D regime; this indicates that both the quasi-2D and 3D flows are highly intermittent. The PIV images provided another indication of the intermittency—both the quasi-2D and 3D flows had coherent vortices which could be distinguished from the background flow. We also applied the γ-test of the hierarchical structure model and found that γ increased from 0.18 for the 3D flow to 0.34 for the quasi-2D flow; the latter value is in accord with expectation for self-similar turbulence. We conclude that our rotating 3D flow is similar to nonrotating turbulent flows, while the rotating quasi-2D turbulence is different from both the 3D rotating turbulence and from nonrotating 2D turbulence studied in other experiments.

ON THE LAW OF MATERIAL FLOW DEFORMATION IN A NONROTATIONAL SYMMETRICAL DRAWING

As a part of systematic study on non-rotational symmetrical drawing, flow displacement and law of materials of blank holder face and case bottom are analyzed according to the measurement of grid strain in rectangular case drawing. Comparing to strain in a cylinderic drawing, the greatest differences include: the compressive strain of circumferential direction in a flange is not proportional to its tensile strain of radial direction, and flow strain in a corner section is non-continuous, and the plastic flow opposite direction of drawing loaded is produced at a short axis of case bottom and other strain circum- stances. All these results are key points, which should be undersood and mastered, to do the calculation with the finite element method. Furthermore, these results also indicate that the forming limit of drawing for larger cover parts could be improved if local lubrication is suitably and reasonably carried out according to the law of materials flow.

Large-eddy simulations of turbulent heat transfer in stationary and rotating square ducts

Turbulent heat transfer at low Reynolds numbers in stationary and rotating straight square ducts was simulated using the large-eddy simulation technique. For the rotating square duct flows, the rotation axis was parallel to two opposite walls of the duct. The pressure-driven flow was assumed to be hydrodynamically and thermally fully developed. A constant and uniform heat flux distribution in the axial direction was applied at the four smooth walls of the forced and mixed convection flows. Two boundary conditions for the thermal energy equation were examined: constant peripheral wall temperature and uniform peripheral wall heat flux. Computations were carried out using a second-order finite volume code with a localized one-equation dynamic subgrid scale model. For the simulated nonrotating flows, there were significant differences of about 16% in the overall Nusselt numbers, depending on the type of thermal boundary condition applied at the walls. This was because of the presence of the corners. Simulations of mixed convection arising from the centrifugal buoyancy effect showed that the turbulence level of the flow was strongly influenced by the centrifugal buoyancy effect. At the rotation rate considered, the overall Nusselt numbers and the turbulence levels of the flow were not greatly affected by the type of boundary conditions for temperature applied at the walls. However, temperature fluctuation intensities noticeably increased when a uniform peripheral heat flux was imposed at the walls.

Reynolds Stress Budgets in Couette and Boundary Layer Flows

Reynolds stress budgets for both Couette and boundary layer flows are evaluated and presented. Data are taken from direct numerical simulations of rotating and non-rotating plane turbulent Couette flow and turbulent boundary layer with and without adverse pressure gradient. Comparison of the total shear stress for the two types of flows suggests that the Couette case may be regarded as the high Reynolds number limit for the boundary layer flow close to the wall. The limit values of turbulence statistics close to the wall for the boundary layer for increasing Reynolds number approach the corresponding Couette flow values. The direction of rotation is chosen so that it has a stabilizing effect, whereas the adverse pressure gradient is destabilizing. The pressure-strain rate tensor in the Couette flow case is presented for a split into slow, rapid and Stokes terms. Most of the influence from rotation is located to the region close to the wall, and both the slow and rapid parts are affected. The anisotropy for the boundary layer decreases for higher Reynolds number, reflecting the larger separation of scales, and becomes close to that for Couette flow. The adverse pressure gradient has a strong weakening effect on the anisotropy. All of the data presented here are available on the web [36].

Three-Dimensional Effects in High-Drag-State Flows over Long Ridges

Abstract Numerical simulations of nonrotating flow with uniform basic wind and stability past long three-dimensional (3D) ridges are compared to the corresponding two-dimensional (2D) limit to reveal the importance of 3D effects. For mountain heights smaller than the threshold for breaking waves, the low-level flow over the interior of the ridge is well described by 2D theory when the horizontal aspect ratio β is roughly 10 or greater. By contrast, in flows with wave breaking significant discrepancies between 2D and 3D results remain apparent even for β = 12. It is found that the onset of wave breaking and the transition to the high-drag state is accompanied in 3D by an abrupt increase in deflection of the low-level flow around the ridge. The increased flow deflection is produced at least in part by upstream-propagating columnar disturbances forced by the transition to the high-drag state. The deflection of the incident flow reduces the amplitude of the mountain wave aloft relative to 2D and acts as a neg...

Improvement of a Low-Reynolds-Number Second Moment Closure and Its Application to Rotating Turbulent Channel Flows.

A low-Reynolds-number second moment closure is improved with the aid of DNS data and applied to turbulent channel flows with a spanwise rotation. Shima’s pressure strain model is refined and a multiple source model is introduced for the exact e equation. Computational results for the turbulent plane channel flows rotating in orthogonal mode are compared with the DNS, LES and experimental data. The present second moment closure achieves a level of agreement with DNS data similar to those for the non-rotating flow cases. In particular, it yields the improved performance in predicting the distributions of e in the near wall sublayer.

Effects of shear and sharp gradients in static stability on two-dimensional flow over an isolated mountain ridge

¶We have investigated the effects of shear and sharp gradients in static stability and demonstrated how a mountain wave and its associated surface winds can be strongly influenced. Linear theory for two-dimensional, nonrotating stratified flow over an isolated mountain ridge with positive shear and constant static stability shows that the horizontal wind speeds on both the lee and upslope surfaces are suppressed by positive shear. The critical F(=U/Nh where U is the basic wind speed, N the Brunt-Vaisala frequency, and h the mountain height) for the occurrence of wave breaking decreases when the strength of the positive shear increases, while the location for the wave-induced critical level is higher in cases with larger positive shear. The linear theory is then verified by a series of systematic nonlinear numerical experiments. Four different flow regimes are found for positive shear flow over a two-dimensional mountain. The values of critical F which separate the flow regimes are lower when the strength of the positive shear is larger. The location of stagnation aloft from numerical simulations is found to be quite consistent with those predicted by linear theory.

Vorticity Budget of Weak Thermal Convection in Keplerian Disks

By employing the equations of mean-square vorticity (enstrophy) fluctuations in strong shear flows, we demonstrate that, unlike energy production of turbulent vorticity in nonrotating shear flows, the turbulent vorticity of weak convection in Keplerian disks cannot gain energy from vortex stretching/tilting by background shear unless the associated Reynolds stresses are negative. This is because the epicyclic motion is an energy sink of the radial component of mean-square turbulent vorticity in Keplerian disks when Reynolds stresses are positive. Consequently, weak convection cannot be self-sustained in Keplerian flows. This agrees with the results implied from the equations of mean-square velocity fluctuations in strong shear flows. Our analysis also sheds light on the explanation of the simulation result in which positive kinetic helicity is produced by the Balbus-Hawley instability in a vertically stratified Keplerian disk. We also comment on the possibility of outward angular momentum transport by strong convection based on azimuthal pressure perturbations and directions of energy cascade.

Effects of Critical Levels on Two-Dimensional Back-Sheared Flow over an Isolated Mountain Ridge on anfPlane

It is well known that there exist three singular points, U 2 c 56 f/k and U 5 c, where the corresponding levels will be called inertia critical levels (ICLs) and the classic critical level (CCL), in the equation governing a two-dimensional, rotating, continuously stratified, hydrostatic, back-sheared Boussinesq flow. Here U and c are basic wind and phase speed, respectively; f is the Coriolis force; and k is the wavenumber. The effects of these critical levels on flows over an isolated mountain ridge are investigated both analytically and numerically, based on a broad range of Rossby numbers (Ro) and Richardson numbers (Ri). Each wave mode generated from the isolated mountain with a continuous spectrum has its own ICL. To indicate the net effects of all ICLs, the authors define the effective ICL as the height above which the amplitude of the inertia-gravity wave mode is very small. The findings for linear flows are summarized as follows. Regime I is inertia-gravity waves. The flow behaves like unsheared inertia-gravity waves and the effective lower ICL plays a similar role as the CCL does in a nonrotating flow. Regime II is combined inertia-gravity waves and baroclinic lee waves. These waves behave like those in regime I below the lower effective ICL, and like baroclinic lee waves near the CCL. In this regime, the horizontal warm advection by the baroclinic lee wave plays an important role in the formation of the lee pressure trough. On the other hand, near the downslope of the mountain, both the warm advection by the inertia gravity and the adiabatic warming also contribute significantly to the lee trough. Therefore, Smith’s quasigeostrophic theory of lee cyclogenesis is extended to a nongeostrophic regime. Regime III is combined evanescent and baroclinic lee waves. These waves still behave like baroclinic lee waves near the CCL, but they are trapped near the surface. Regime IV is transient waves. Nongeostrophic baroclinic instability exists, as evidenced by the positive domain-averaged north‐south heat flux. There exists no steady state. At earlier times, the flow behaves like trapped baroclinic lee waves. For a relatively large Ri (e.g., Ri . 25), the flow falls into regime I when Ro is relatively large (e.g., Ro $ 2). It then shifts to regime II, and finally to regime III as Ro decreases. For a relatively small Ri (e.g., Ri , 25), the flow shifts from regime I to II, and then finally to regime IV when both Ro and Ri decrease.

Wave Ducting in a Stratified Shear Flow over a Two-Dimensional Mountain. Part II: Implications for the Development of High-Drag States for Severe Downslope Windstorms

Abstract In this study, it is found that the discrepancies among earlier studies of severe downslope windstorms are caused by the use of the critical level height (zc), instead of the low-level uniform flow–layer depth (z1), as an indicator to determine the optimal conditions for the occurrence of high-drag states. It is determined that once the wave breaking occurs, it induces a critical level and establishes a flow configuration favorable for wave ducting in the lower uniform wind layer, which determines the phase of reflected waves. Flow regimes of high- and low-drag states for a two-dimensional, nonrotating flow with uniform static stability and a basic-state critical level over a mountain were also determined as functions of nondimensional mountain height (h), Richardson number (Ri), and nondimensional z1 in the terrain-following coordinates (σ1). The authors found that 1) the critical h for high-drag state increases as Ri increases when σ1 is fixed, 2) the critical h for high-drag state increas...