Laboratory and numerical experiments on the near wake of a sphere in a stably stratified ambient

The flow around and behind a sphere in a linear density gradient has served as a model problem for both body-generated wakes in atmospheres and oceans, and as a means of generating a patch of turbulence that then decays in a stratified ambient. Here, experiments and numerical simulations are conducted for 20 values of Reynolds number, $Re$ , and internal Froude number, $Fr$ , where each is varied independently. In all cases, the early wake is affected by the background density gradient, notably in the form of the body-generated lee waves. Mean and fluctuating quantities do not reach similar states, and their subsequent evolution would not be collapsible under any universal scaling. There are five distinguishable flow regimes, which mostly overlap with previous literature based on qualitative visualisations and, in this parameter space, they maintain their distinguishing features up to and including buoyancy times of 20. The possible relation of the low $\{Re, Fr\}$ flows to their higher $\{Re, Fr\}$ counterparts is discussed.

An unambiguous definition of the Froude number for lee waves in the deep ocean

There is a long-standing debate in the literature of stratified flows over topography concerning the correct dimensionless number to refer to as a Froude number. Common definitions using external quantities of the flow include $U/(ND)$, $U/(Nh_{0})$, and $Uk/N$, where $U$ and $N$ are, respectively, scales for the background velocity and buoyancy frequency, $D$ is the depth, and $h_{0}$ and $k^{-1}$ are, respectively, height and width scales of the topography. It is also possible to define an internal Froude number $Fr_{\unicode[STIX]{x1D6FF}}=u_{0}/\sqrt{g^{\prime }\unicode[STIX]{x1D6FF}}$, where $u_{0}$, $g^{\prime }$, and $\unicode[STIX]{x1D6FF}$ are, respectively, the characteristic velocity, reduced gravity, and vertical length scale of the perturbation above the topography. For the case of hydrostatic lee waves in a deep ocean, both $U/(ND)$ and $Uk/N$ are insignificantly small, rendering the dimensionless number $Nh_{0}/U$ the only relevant dynamical parameter. However, although it appears to be an inverse Froude number, such an interpretation is incorrect. By non-dimensionalizing the stratified Euler equations describing the flow of an infinitely deep fluid over topography, we show that $Nh_{0}/U$ is in fact the square of the internal Froude number because it can identically be written in terms of the inner variables, $Fr_{\unicode[STIX]{x1D6FF}}^{2}=Nh_{0}/U=u_{0}^{2}/(g^{\prime }\unicode[STIX]{x1D6FF})$. Our scaling also identifies $Nh_{0}/U$ as the ratio of the vertical velocity scale within the lee wave to the group velocity of the lee wave, which we term the vertical Froude number, $Fr_{vert}=Nh_{0}/U=w_{0}/c_{g}$. To encapsulate such behaviour, we suggest referring to $Nh_{0}/U$ as the lee-wave Froude number, $Fr_{lee}$.

Exchange flow between a canopy and open water

This paper theoretically and experimentally investigates the exchange flow due to temperature differences between open water and a canopy of aquatic plants. A numerical model is used to study the interfacial shape, frontal velocity and total volumetric exchange, and their dependence on a dimensionless vegetation drag parameter. The numerical predictions are consistent with the laboratory measurements. There is a short initial period in which the force balance is between buoyancy and inertia, followed by drag-dominated flow for which there is a balance between buoyancy and drag forces. After the initial stage, the gravity current propagating into the canopy takes a triangular shape whereas the current propagating into the open water has almost the classic unobstructed horizontal profile, but with a slowly decreasing depth. Near the edge of the canopy, but in the open region, the flow is found to be critical with a unit internal Froude number. The exchange flow rate and the front speed in the canopy decrease slowly with time whereas the gravity current in the open water has a constant speed. The magnitude of the exchange flow decreases as the canopy drag increases. Empirical equations for the flow properties are presented. A movie is available with the online version of the paper.

The evolution of initially turbulent bluff-body wakes at high internal Froude number

Coherent vortex structures are formed in the late wakes of towed spheres for all values of the internal Froude number, F≡2U/ND∈ [10, 240] (U is the body speed, D its diameter, and N is the buoyancy frequency). The eventual emergence of the long-lived and stable pattern of alternating-signed patches of vertical vorticity is characteristic of all towed-sphere wakes, from those dominated by internal lee waves at F=1, to initially fully turbulent early wakes at F[ges ]4. At late times, the local Froude number is always low, and a characteristic stratified wake structure and dynamics result. These wakes have high mean wake defect velocities compared with non-stratified wakes, but the decay rates of energy and enstrophy are similar. Experimental evidence is presented for the existence of an intermediate non-equilibrium (NEQ) regime with very low decay rates of kinetic energy, that precedes the late wake. The NEQ regime is the period when the initial turbulence reorganizes under the increasingly (relative to the decaying turbulent kinetic energy) powerful influence of the background density gradient, accompanied by conversion of potential to kinetic energy as the wake turbulence collapses. The stable long-lived late-wake structure that eventually emerges has a high degree of order and coherence that reflects the initial wake instability. A universal curve for the energy decay of all stratified drag wakes at high Froude and Reynolds numbers is proposed.

Flows generated by the periodic horizontal oscillations of a sphere in a linearly stratified fluid

The flow field induced by a sphere oscillating horizontally in a linearly stratified fluid is studied using a series of laboratory experiments. The resulting flows are shown to depend on the Stokes number β, the Keulegan–Carpenter number KC and the internal Froude number Fr. For Fr [clubs ] 0.2, it is shown that the nature of the resulting flow field is approximately independent of Fr and, based on this observation, a flow regime diagram is developed in the (β, KC)-plane. The flow regimes include: (i) fully-attached flow; (ii) attached vortices; (iii) local vortex shedding; and (iv) standing eddy pair. An internal-wave flow regime is also identified but, for such flows, the motion field is a function of Fr as well. Some quantitative measures are given to allow for future comparisons of the present results with analytical and/or numerical models. Wherever possible, the results are compared with the experiments of Tatsuno & Bearman (1990) on right circular cylinders oscillating in homogeneous fluids.

Linearly stratified flow past a horizontal circular cylinder

The flow of a linearly stratified fluid past a long circular cylinder in a channel is considered experimentally. The characteristics of the flow depend on the internal Froude number F i the Reynolds number Re and the cylinder diameter to fluid depth ratio, d/H . A wide range of characteristic flow fields are observed in the parameter space investigated; i.e. 0.02 ⩽ F i ⩽ 13 , 5 ⩽ Re ⩽ 4000 and 0.03 ⩽ d / H ⩽ 0.20 . A flow regime diagram of F i against Re for these characteristic flows is developed. Some of the lower F i Re experiments are compared with numerical experiments. A theory is advanced which indicates that the dimensionless length, x b ∗ = x b / d of the blocked region ahead of the cylinder should scale as x b ∗ ≈ ( δ / d ) 5 R e F i − 2 , where δ is the thickness of the shear layer between the external flow and the approximately stagnant blocked region; the results of an experimental programme that support this scaling are presented. Measurements are made which indicate that for the range of parameter space in which lee waves occur, the lee wavelengths are predicted to a good approximation by linear theory. A scaling analysis is carried out which suggests that the height of the rotors above the streamwise centreline, Z r ∗ = Z r / d , scales with F i experiments aire in good agreement with this prediction. For conditions under which the wake of the cylinder is turbulent, scaling arguments suggest that the dimensionless maximum width of the wake, γ m ∗ = γ m / d , and the dimensionless streamwise distance at which this maximum occurs, β m ∗ = β m / d , scale as F i 1 2 Experiments are presented which support this scaling.