Infinite Froude Number Research Articles

Static hydroelastic study of composite T-foils with beam and lifting line models

Well-designed hydrofoils improve ship resistance and seakeeping by lifting the hull above the water. With greater speeds come greater loads, and the two-way interaction of structural deflections of lifting surfaces on the hydrodynamics must be considered. Tailored structural anisotropy can improve hydrodynamic and structural efficiency of lifting surfaces compared to rigid counterparts by exploiting the layup of composite materials. Structural efficiency here means reduced risk of structural failure for a given amount of material, and hydrodynamic efficiency means lower drag. A T-foil is a prototypical multi-component appendage, consisting of the foil wing and the strut, which we investigate in this work. We use simple composite beam and lifting line theory to explore the static fluid-structure interaction of a composite T-foil for a variety of fiber angles (θf = 0˝ , ˘15˝ ). We apply a simple approximation on lift coefficient at infinite Froude number (F n) to model the free-surface effects, which is valid at high depth-based Froude numbers (F nh ą10a h/c) when CL is independent of F nh and the inertial effects dominate. Results for the moth rudder T-foil geometry studied here indicate that aligning composite fibers towards the leading edge results in a more hydrostructurally efficient foil and that free-surface effects are minor because of the large submergence for this flow condition.

Froude Number Effects on Two-Dimensional Hydrofoils

The performance of a two-dimensional hydrofoil of arbitrary camber, moving at arbitrary Froude number at a constant depth below a free surface, is considered. The treatment is based upon the use of singularity distributions and thin foil theory. By assuming an appropriate series form for the vortex distribution representing the hydrofoil, it is shown that the problem can be reduced to the solution of a set of linear algebraic equations. These are solved by a collocation procedure. Numerical results for the performance characteristics are then given for several hydrofoil configurations, submergence depths, and Froude numbers. These indicate that operation at Froude numbers greater than about ten is practically equivalent to operation at infinite Froude number. However, at lower values of the Froude number and for all the configurations considered, Froude number effects are important, even at submergence depths of several chord lengths.

Dense underflow into a lake or reservoir – supercritical flow solutions

Steady two-dimensional flow from an angled structure into a lake or a reservoir where the interface between the intrusion and the ambient fluid separates from a solid wall is considered. The fluid is assumed to be of finite depth and the incoming channel makes a downward angle α with the horizontal axis. This simple configuration provides a model for the plunging inflow and subsequent underflow of dense water in a reservoir or lake. Exact solutions are presented at infinite Froude number and compared with the solutions to the full nonlinear problem for supercritical flow. Limiting flows are found to separate from the upper boundary at a stagnation point, and regions of non-uniqueness in the solution domain are found.

Stability of Viscous St. Venant Roll Waves: From Onset to Infinite Froude Number Limit

We study the spectral stability of roll-wave solutions of the viscous St. Venant equations modeling inclined shallow-water flow, both at onset in the small-Froude number or "weakly unstable" limit $F\to 2^+$ and for general values of the Froude number $F$, including the limit $F\to +\infty$. In the former, $F\to 2^+$, limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV) equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate this formal limit, showing that stability as $F\to 2^+$ is equivalent to stability of the corresponding KdV-KS waves in the KdV limit. Together with recent results obtained for KdV-KS by Johnson--Noble--Rodrigues--Zumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St. Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainder of the paper, we investigate numerically and analytically the evolution of the stability diagram as Froude number increases to infinity. Notably, we find transition at around $F=2.3$ from weakly unstable to different, large-$F$ behavior, with stability determined by simple power law relations. The latter stability criteria are potentially useful in hydraulic engineering applications, for which typically $2.5\leq F\leq 6.0$.

Numerical study on horizontal convection of a rarefied gas over a non-isothermal plane wall

A rarefied gas over an infinite plane wall with non-uniform periodic temperature distribution is considered under the effect of gravity. The Knudsen number and the Froude number are defined as the mean free path of gas molecules and the scale height at a reference state divided by the length of the period, respectively. Based on the kinetic theory of gases, the steady two-dimensional gas flow is investigated numerically for a wide range of parameters. The cases of a free molecular gas are analyzed by a deterministically accurate method, which enables the computation for large Froude numbers, i.e., vanishingly small gravity. The flow pattern is shown to be slightly effected by the Froude number when the Froude number is large, whereas the flow magnitude is proportional to the inverse of the Froude number. As a result, the flow vanishes in the limit of zero gravity. This is not a trivial consequence because the case of an infinite Froude number is different from the same setting without gravity. The cases of finite Knudsen numbers are investigated by the direct simulation Monte Carlo method for a hard sphere gas, and the flow characteristics are shown to be dominated by the presence of gravity for cases in which the Knudsen number is larger than the Froude number.

Nonlinear Computation of Water Impact of Axisymmetric Bodies

A fully nonlinear numerical simulation based on a boundary element method was used to investigate water impact of axisymmetric bodies that strike vertically the horizontal free surface from the air. The main objective was to understand the gravity effect on flow/wave kinematics and dynamics and to quantify the range of validity of existing theories and computations that are based on the infinite Froude number assumption. Two body geometries were considered: inverted cone and sphere. For the inverted cone, we obtained detailed dependencies of free-surface profile and impact pressure and load on the body on the generalized Froude number (Fr(V/gt)1/2, where V is the impact velocity, g is the gravitational acceleration, and t is time) and deadrise angle a. Based on these, we developed an approximate formula for evaluating the contribution of the gravity effect to the total impact force on the body in terms of a similarity parameter Fr/a1/2. For the sphere, we developed and applied a pressure-based criterion to follow the evolution of flow separation on the body and to obtain an appropriate description of the free-surface profile near the body and accurate evaluation of the impact pressure and load on the body during the entire impact process. The numerical result of impact force on the body agreed well with existing experimental measurements. We confirmed that the gravity effect is unimportant in initial impact of the sphere. Significantly, we found that in a later stage of impact, flow separation remains at an almost fixed position at an angle u 62.5 deg to the bottom of the sphere for a wide range of Froude numbers, Fr V/(gR)1/2 1, where R is the radius of the sphere.

The merger of two-dimensional radially stratified high-Froude-number vortices

We investigate the influence of density inhomogeneities on the merger of two corotating two-dimensional vortices at infinite Froude number. In this situation, buoyancy effects are negligible, yet density variations still affect the flow by pure inertial effects through the baroclinic torque. We first re-address the effects of a finite Reynolds number on the interaction between two identical Gaussian vortices. Then, by means of direct numerical simulations, we show that vortices transporting light fluid in a heavier counterpart merge from further distances than vortices in a uniform density medium. On the other hand, heavy vortices only merge from small separation distances. We measure the critical distance a/b0 of the vortex radii to their initial separation distance. It departs from the homogeneous threshold of 0.22 in response to increasing density contrasts between the vortices and their surroundings. An analysis of the contribution of the baroclinic vorticity to the dynamics of the flow is detailed and explains the observed behaviour. This analysis is completed by a simple model based on point vortices that mimics the flow. It is concluded that vortices carrying light fluid are more likely to generate large-scale structures than heavy ones in an inhomogeneous fluid.

Analysis of supercavitating and surface-piercing propeller flows via BEM

A low-order potential based 3-D boundary element method (BEM) is presented for the analysis of unsteady sheet cavitation on supercavitating and surface-piercing propellers. The method has been developed in the past for the prediction of unsteady sheet cavitation for conventional propellers. To allow for the treatment of supercavitating propellers, the method is extended to model the separated flow behind trailing edge with non-zero thickness. For surface-piercing propellers, the negative image method is used, which applies the linearized free surface boundary condition with the infinite Froude number assumption. The method is shown to converge quickly with grid size and time step size. The predicted cavity planforms and propeller loadings also compare well with experimental observations and measurements.

Bow flows with smooth separation in water of finite depth

AbstractThe bow flow generated by a wide flat-bottomed ship moving in water of finite depth is examined. Solutions obtained using an integral equation technique are presented for a range of different depths and for a range of angles of the front of the bow. The solution for the limiting case of infinite Froude number is obtained as an integral, and numerical solutions are found for the nonlinear problem in which the Froude number is finite. Solutions with smooth separation are shown to exist for all values of Froude number greater than unity, for any bow slope.

Flow from a vertical slot into a layer of finite depth

The flow of water upward from a vertical slot into a fluid of finite depth is considered. The shape of the free surface is computed for a range of parameter values. It is shown that solutions in which the free surface separates smoothly from the vertical wall can be obtained for a range of Froude numbers and slot sizes but that the absolute lower bound on these flows is F = 1, provided the slot width is less than approximately one half the depth of the layer. If the slot size is larger than this, the lower bound on F increases. Solutions for infinite Froude number are shown toexist up to a maximum slot width equal to the depth of the flowing layer, and for finite Froude number a limiting value on the slot width is found that depends upon the Froude number.

A note on the free surface induced by a submerged source at infinite Froude number

AbstractThe free surface due to a submerged source in a fluid of finite depth at infinite Froude number is reconsidered. A conformal transformation technique is used to formulate this problem as an integral equation for the free-surface angle. An elementary solution is found for the equation, which results in a closed form expression for the free-surface elevation. Comparison is made with previous numerical solutions.

Absolute instability in hot jets

The linear inviscid stability of heated compressible axisymmetric jets is re-examined, focusing on the impulse response of the flow. The case is of particular interest in which a transient grows exponentially in place, i.e., at the location at which it was generated. Such a situation is commonly termed absolutely unstable as opposed to correctively unstable. Limiting the analysis to locally parallel mean velocity and density profiles, infinite Froude number, and zero Eckert number, the absolute instability boundaries, i.e., the boundaries between absolute and convective instability, are explored in the mean flow parameter space. The influence of mean flow profiles, exit Mach number, exit temperature ratio, and exit velocity ratio are considered and discussed separately. It is shown that heated jets, i.e, jets with lower than ambient density, can, depending on operating conditions, develop a region of local absolute instability in the potential core region if the jet density is less than 0.72 times the ambient density. The results suggest that a self-excited global resonance may dominate the heated jet response in situations of practical interest.

Infinite Froude number solutions to the problem of a submerged source or sink

AbstractThe problem of a source or sink submerged beneath a free surface is investigated in the infinite Froude number limit. Solutions are found for all cases in which the source is situated away from the bottom of the channel. Solutions are also found for the case where the source is situated above the asymptotic level of the free surface, giving fountain type free surface shapes.

A theoretical investigation of an asymmetric planing hull at infinite Froude number

AbstractThe extent to which an asymmetric low-aspect-ratio flat ship is wetted when planing at infinite Froude number is investigated, with emphasis placed on its relationship with the shape of the hull. Two cases are considered. First the hull is assumed to have two laterally-asymmetric leading edges and, secondly, the hull is assumed to be yawed sufficiently for one of the leading edges to become a trailing edge. In the first case, the relationship involves a pair of coupled integral equations, but in the second case there is a complication by the occurrence of hull-wake interaction.

Calculation of the wetted area of a planing hull with a chine

The extent to which a low-aspect-ratio flat ship with a chined hull is wetted when planing at infinite Froude number is investigated. A numerical method of solution for the wetted area, which is applicable to more general planing problems, is presented. The results obtained by this method are compared with those found by solving the inverse problem of determining the hull shape which produces a given waterplane shape and are shown to be in excellent agreement. Results are also presented which indicate that a ‘vertical’ chine may be used to fix the shape of the wetted region.

Planing of a low-aspect-ratio flat ship at infinite Froude number

The free-surface elevation caused by the planing motion of a low-aspect-ratio flat ship at infinite Froude number is investigated. A relation is obtained between the physical characteristics of the planing hull and the extent to which it is wetted. Analytic results are presented in the case of laterally-uniform longitudinal hull slope.