Bernoulli Function Research Articles

Solvability of a three-dimensional stationary flowing problem

where v is the velocity field of the fluid, H ≡ P +v2/2 is the Bernoulli function, and P is the pressure. We suppose that the flow domain Ω ⊂ R3 is bounded and fixed. Denote the boundary of Ω by Γ. The normal component γ of v on Γ gives rise to the partition of Γ into three disjoint sets: the rigid wall Γ0, the inlet part Γ+ through which the fluid flows into the domain, and the outlet part Γ− through which the fluid flows out the domain: Γ0 = {x ∈ Γ : γ(x) = 0}, Γ+ = {x ∈ Γ : γ(x) 0}. The set Γ+ is referred to as the inlet. If γ 6≡ 0 then we have the flowing problem for Ω. Without additional boundary conditions, a solution to the problem is nonunique and depends on arbitrary functions. In the present article, we study the stationary flowing problem (SFP) that comprises the Euler equations (1), the circulation equations ∮

An unsteady extension of Bernoulli's theorem.

An unsteady extension of Bernoulli's theorem is presented. For steady flows, the derivative of the Bernoulli function in the normal direction perpendicular both to the streamline and the gradient of potential temperature becomes equal to the product of the flow speed and the binormal component of absolute vorticity. Even for unsteady flows, the same formula holds except that the absolute vorticity is replaced by the sum of the absolute vorticity and the local rotation rate of velocity.

Properties of similarity solutions for flows in turbulent vortex cores

A class of similarity solutions of the equations for turbulent vortex cores matching an external inviscid similarity flow with a power law of circumferential velocity variationv-r −m near the rotation axis and constant Bernoulli function is considered. Solutions are found to exist only in a certain range of the indexm of the exponential. For each suchm there are two solutions.

Certain features of similarity solutions for flows in viscous vortex cores

A class of steady similarity solutions of the equations for viscous vortex cores which correspond to external inviscid similarity solutions with a power-law variation of the circumferential velocityv-r −m near the rotation axis is considered. It is found that if the Bernoulli function in external flow is constant, then these solutions will exist only on a certain range of the indexm of the exponential. For eachm on this range there are two solutions.

Sufficient stability criteria for fluids and plasmas

The stability of fluids and multifluid plasma equilibria is discussed in close analogy to classical mechanics: Two canonical transformations (involving the equilibrium Bernoulli function) lead to a new Hamiltonian which has the property of a general Lyapunov functional: It is constant in time within the perfect-fluid equations, its first variation is zero for a general equilibrium, and its finite second variation has a definite sign if certain inequalities are fulfilled, implying nonlinear stability.

The Pacific Subsurface Countercurrents and an Inertial Model*

The Tsuchiya jets, or subsurface countercurrents, extend across the Pacific Ocean carrying 7 (±2) × 106 m3 s−1 eastward on each side of the equator. Mean meridional sections of potential temperature, salinity, neutral density anomaly, and the square of buoyancy frequency are presented for the western, central, and eastern tropical Pacific Ocean. These sections are used together with maps of depth and salinity on isopycnals, as well as thickness between isopycnals, to describe the evolution of the Tsuchiya jets as they flow from west to east. An inertial-jet model is formulated in which conservation of the Bernoulli function and potential vorticity combine with the eastward shoaling of the tropical pycnocline to dictate the jet structure. This model jet is consistent with a number of features of the Tsuchiya jets: their roughly constant volume transports, their advection of properties such as salinity and oxygen over long zonal distances, their rapidity and narrowness, their poleward shift from west to east, the large potential vorticity gradients across them, and the pycnostad between them that builds in size and strength from west to east. However, an observed decrease in density carried by the Tsuchiya jets from west to east, not included in the model jet, suggests that diffusive or advective interaction with the surrounding ocean also may be important in subsurface countercurrent dynamics.

Momentum and Energy Balance of the Mediterranean Outflow

Field data taken in the Gulf of Cadiz have been analyzed to describe some aspects of the momentum and energy balance of the Mediterranean outflow. A crucial component of the momentum balance is the total stress (entrainment stress and bottom drag), which has been estimated from a form of the Bernoulli function evaluated from density and current observations. For the first 60 km west of the Camarinal Sill the outflow was confined within a narrow channel on the continental shelf. At about 70 km downstream the outflow crossed over the shelf‐slope break and began to descend the continental slope. The buoyancy force increased substantially, and the outflow underwent a geostrophic adjustment, albeit one heavily influenced by mixing and dissipation. The current direction turned 90 degrees to the right at a near-inertial rate. In this region, the estimated geostrophic velocity greatly underestimated the actual current, and the estimated curvature Rossby number was about 0.5. Current speeds were in excess of 1ms 2 1 and the total stress was as large as 5 Pa. The entrainment stress, estimated independently from property fluxes, reached a maximum of about 1 Pa, or considerably smaller than the inferred bottom stress. By about 130 km downstream, the current was aligned approximately along the local topography. The current amplitude and the estimated stress were then much less, about 0.3 m s21 and 0.3 Pa. The entrainment stress was also very small in this region well downstream of the strait. This slightly damped geostrophic flow continued on to Cape St. Vincent where the outflow began to separate from the bottom. Bottom stress thus appears to be a crucial element in the dynamics of the Mediterranean outflow, allowing or causing the outflow to descend more than a kilometer into the North Atlantic. In the regions of strongest bottom stress the inferred drag coefficient was about 2 2 12 (3 10 23) depending upon which outflow speed is used in the usual quadratic form. Entrainment stress was small by comparison to the bottom stress, but the entrainment effect upon the density anomaly was crucial in eroding the density anomaly of the outflow. The observed entrainment rate appears to follow, roughly, a critical internal Froude number function.

Lyapunov stability of perfect fluid and multifluid plasma equilibria

A Lyapunov functional F for neutral fluids and for multifluid plasmas is found by a canonical transformation of the total Hamiltonian; this transformation is motivated by the requirement that all canonical variables (including Clebsch variables) should be time-independent in a perfect-fluid equilibrium. Then we find δF = 0 for such an equilibrium, and δ 2F ≥ δ 2F ∗ > 0 if the local flow velocity and the Bernoulli function of each fluid component obey certain inequalities (sufficient stability conditions). Implications for dissipative equilibria are also discussed.

F平面上での2次元定常な流れに於ける保存量と山を越える流れへの応用

A steady-flow system on an f-plane whose velocity is uniform in one horizontal direction, say y direction, is considered. The velocity far upstream in the other horizontal direction, say minus x direction, is assumed to be horizontal and dependent only on the vertical coordinate, say z. The potential temperature, Bernoulli function or potential vorticity is not necessarily conserved along the stream lines projected onto the 2-dimensional (z, x) vertical plane, although they are conserved along the true stream lines in the 3-dimensional (z, x, y) space. In this note, a temperature-like quantity Θ, an energy-like quantity B and a vorticity-like quantity Q, which are conserved along the projected stream lines on the (z, x) vertical plane, are constructed. They become functions of the 2-dimensional stream function ψ=ψ(z, x). As in the case of the true Bernoulli function and potential vorticity, Q=Q[ψ] is the derivative of B=B[ψ], i. e., Q[ψ]=dB[ψ]/dψ.As an application of the conservation laws, a problem of hydrostatic neutral flow with a horizontal rigid lid over a mountain is exactly solved. In particular, both the z and x components of velocity are shown not to be affected by the rotation.

On the branching of the Kuroshio west of Kyushu

Trajectories of satellite‐tracked surface drifters released in the continental margin area west of Kyushu indicate a separation of the inshore side of the Kuroshio that appears to form the beginning of the Tsushima Current. The trajectories also indicate an anticyclonic sense of circulation in the area between the separated branch and the eastward directed Kuroshio main stream, consistent with long‐term geomagnetic electrokinetograph measurements of surface currents, which show a broad southward flow along the west Kyushu coast. A theory is put forth that ties together these observed flow features. In the theory, the incidence is considered of an inertial current upon a step rise in topography (characterized by a step depth of H0 to simulate the west Kyushu coastal bathymetry) in an f plane, two‐layered, inviscid ocean with a quiescent lower layer. The inertial current is of a constant potential vorticity f/H, where H (>H0) is the undisturbed upper layer depth and is bounded inshore by a free streamline along which the upper layer depth is a constant hc (<H). The steady state downstream configuration of a separation branch and a deflected main stream along the step is connected to the upstream approach current through the consideration of the constancy of the Bernoulli function and the conservation of potential vorticity and mass. In the immediate neighborhood of the incidence, a complicated flow region is anticipated in which recirculation is featured and an anticyclonic gyre is expected further onto the step. The volume‐integrated balance of along‐step momentum of the constant potential‐vorticity flow allows the determination of the angle the branch current makes with the step as a function of the angle of incidence and two depth ratios, hc/H and H0/H. The bifurcation quotient, defined as the ratio of the branch current transport to that of the approach current, is found to be also such a function, except it is independent of the incidence angle. These functions yield results that are qualitatively in agreement with the observed bifurcation of the Kuroshio west of Kyushu that gives rise to the Tsushima Current.

ベルヌウイ関数と渦位の間の一つの関係式

In 2-dimensional steady adiabatic unforced systems, both the Bernoulli function B and potential vorticity Q are functions of the stream function Ψ, and Q is the derivative of B with respect to Ψ, i.e., Q = dB/dΨ. In this note, this formula is extended to 3-dimensional steady adiabatic systems. Even if there exists dissipation, essentially the same formula holds as far as the inner product between the dissipative force and the velocity is everywhere negative. An application is presented to the problem of lee vorticity.

The Bernoulli function and flux of energy in the ocean

This short note explains how the Bernoulli function, the total energy of a fluid, can be calculated from recent work on the thermodynamics of seawater. The Bernoulli function for an incompressible fluid has been used in the past to infer the absolute circulation of the ocean from hydrographic data alone. This is now both undesirable and unnecessary. Constructing the Bernoulli function for a hydrographic section also provides a rigorous procedure for calculating the total energy flux across the section, a quantity dominated by the “heat flux”.

A Generalization of Bernoulli's Theorem

Abstract The conservation of potential vorticity Q can be expressed as ∂(ρQ/∂t + ∇·J = 0, where J denotes the total flux of potential vorticity. It is shown that J is related under statistically steady conditions to the Bernoulli function B by J = ∇θ × ∇B,where θ is the potential temperature. This relation is valid even in the nonhydrostatic limit and in the presence of arbitrary nonconservative forces (such as internal friction) and heating rates. In essence, it can be interpreted as a generalization of Bernoulli's theorem to the frictional and diabatic regime. The classical Bernoulli theorem—valid for inviscid adiabatic and steady flows—states that the intersections of surfaces of constant potential temperature and constant Bernoulli function yield streamlines. In the presence of frictional and diabatic effects, these intersections yield the flux lines along which potential vorticity is transported.

The intrusion of the Kuroshio across the continental shelf northeast of Taiwan

Hydrographic observations in an area immediately northeast of Taiwan in April 1989 indicate an on‐shelf intrusion of Kuroshio water across a sharply curved continental shelf break. It appears that a part of the Kuroshio on the cyclonic side overran the shelf break and penetrated northward as a shallow surface current. The remainder of the Kuroshio, presumably affected by the shoaling topography, largely turned and ran along the shelf break to the northeast. Between the two, the flow was weak and disorganized. Conservation of potential vorticity and constancy of the Bernoulli function in an analytical, reduced‐gravity model of the incidence of a baroclinic current upon a step shelf lead to an on‐shelf flow field that is compatible to the observed hydrographic distributions. In particular, the integrated balance of along‐step momentum yields an expression of the angle of intrusion in terms of the incidence angle and of the ratio of step depth to the depth of the upper layer of the ocean. In addition, the transport of the intrusion is shown to be equal to the product of the depth ratio squared and the incident transport. Calculations for the condition of the April survey yield results in agreement with the observation.

Finite-Element Method for Three-Dimensional Incompressible Viscous Flow Using Simultaneous Relaxation of Velocity and Bernoulli Function. 2nd Report. Natural Convection in Horizontal Concentric Annuli.

In previous research, a simple finite-element method using simultaneous relaxation of velocity and the Bernoulli function was presented to simulate three-dimensional flow prblems. In the present paper, this scheme is expanded to consider heat convection problems. In order to reduce the computational storage, coefficient matrices are calculated by simple integration formulas for an eightnode isoparametric element. Natural convection in horizontal concentric annuli is calculated to certify the scheme. The present method is stable in heat convection analysis and the numerical results agree well with the experimental results.

Finite-Element Method for Three-Dimensional Incompressible Viscous Flow Using Simultaneous Relaxation of Velocity and Bernoulli Function : Flow in a Lid-Driven Cubic Cavity at Re=5000

Finite-Element Method for Three-Dimensional Incompressible Viscous Flow Using Simultaneous Relaxation of Velocity and Bernoulli Function : Flow in a Lid-Driven Cubic Cavity at Re=5000

Finite-Element Analysis of Natural Convection in Concentric Horizontal Annuli.

In this paper, a simple FEM scheme is presented which simulates the natural convection of incompressible viscous flow. The Navier-Stokes equations in rotational form with the buoyancy effect is simplified by Boussinesq's approximation. The main algorithm of the GSMAC method is employed to obtain the time=dependent solutions. The equation of continuity is satisfied by the simultaneous relaxation of velocity and Bernoulli function. Natural convections in concentric horizontal annuli at various Rayleigh numbers are numerically investigated in order to validate the present method. The temperature distributions obtained here are in good agreement with the experimental results.

The Link between Western Boundary Currents and Equatorial Undercurrents

A constant potential vorticity model is used to investigate the relation between the properties of an equatorward-directed western boundary current and the formation of the Equatorial Undercurrent (EUC) in the western equatorial oceans. It is shown that the value of the Bernoulli function at the equator, undetermined in earlier purely inertial EUC models, is fixed by the bifurcation latitude of the western boundary current that feeds the EUC. This is an additional example of the nonlocal forcing of the EUC by the subtropical gyre. The constant potential vorticity model yields reasonable values for the initial EUC core velocities for a bifurcation latitude on the order of 6° from the equator.

Finite-Element Method for Three-Dimensional Incompressible Viscous Flow Using Simultaneous Relaxation of Velocity and Bernoulli Function. 1st Report: Flow in a Lid-Driven Cubic Cavity at Re=5000.

A simple finite-element method for incompressible viscous flow is presented. This method is a simplified version of the GSMAC finite-element method, and it is applicable to large systems. Navier-Stokes equations in rotational form and the equation of continuity are employed as the governing equations. A time dependent solution is obtained by the following procedures. (1) Prediction of velocity field by explicit time advancement. (2) Correction of both velocity and Bernoulli function by simultaneous relaxation satisfying the equation of continuity. Unsteady flow in a lid-driven cubic cavity at the Reynolds number of 5000 is numerically investigated to verify the present method. Velocity profiles in the cavity are in good agreement with the experimental results by Prasad. The present method is stable in three-dimensional analysis and non physical pressure oscillations are not observed.

Toroidal equilibria of an electron gas

A general formulation for investigating axisymmetric toroidal equilibria of pure electron clouds is given. The fluid and Maxwell's equations are reduced to a set of three coupled equations for the (covariant) toroidal components of the generalized vorticity ( F) and velocity ( j), and the Bernoulli function ( U). There appear two arbitrary flux functions I 0 and Ψ 0, corresponding to the toroidal magnetic field and to the poloidal flux of the generalized vorticity. Experimentally relevant exact analytical and numerical solutions are presented for appropriate boundary conditions.