Unsteady Potential Flow Research Articles

Improvements to a tightly coupled viscous–inviscid procedure for three-dimensional unsteady transonic flow

A tightly coupled procedure for solving three-dimensional compressible viscous flows has been developed. In this approach, the flow field is divided into two domains, a viscous domain surrounding viscous wakes and solid surfaces such as wings or rotor blades, and an outer domain where the inviscid, unsteady compressible potential flow equations hold. These two zones are tightly coupled to each other, allowing conservation of mass, momentum and energy at the interface, and permitting the propagation of non-linear acoustic waves from one zone to another in a seamless fashion. The location of the interface dividing the two zones is allowed to adjust its position relative to blade based on an investigation of the instantaneous flow physics. The effects of rotor trim, aeroelasticity, and far wake induced inflow are included in the analysis through a coupling with a comprehensive rotor analysis. The methodology has been applied to two cases which represent the flow physics typically encountered by modern helicopter rotors. A cruise condition for a UH-60 is analyzed and the results compared with both experimental data and full Navier–Stokes results. Computer time savings realized by the zonal method over the full Navier–Stokes analysis are given. A non-lifting high speed case is also investigated to assess the accuracy of unsteady shock motion. The shock structure is examined to determine whether inaccuracies are introduced by the presence of the interface.

Random walks and hydrodynamical lift from wing sections

The author presents a numerical method for obtaining the lift coefficient of wing sections by introducing a random walk algorithm. The phenomenon of migrating fluid fronts in filter paper around macroscopic obstacles leads to a numerical model of unsteady potential flow the author calls “random walk source model”. The shape of the migrating fluid fronts simulated in the random walk model tells the history of the path of fluid particles leading them to the front, and thus carrying the information about the lift coefficient of the passed wing section. He studied the migration of simulated fluid fronts around Joukowsky profiles whose lift coefficient is known, and compares the theoretical given lift coefficient with the numerically determined value, resulting from the random walk experiment. Even though the contour of the Joukowsky profile is approximately by only 100 to 200 pixel units, it is possible to predict the lift coefficient with an accuracy of 10%.

Analysis of elastic cavitating wing vibration

A two-dimensional unsteady potential flow outside an elastic partially cavitating wing is analyzed numerically by using the Birnbaum equation on the hydrofoil chord and the Lagrange–Cauchy integral on the cavity. The angle of attack has a small periodic perturbation, and cavity thickness and length also take perturbations. Wing vibration is considered to be vibration of a variable thickness beam with two clamp bolts near the beam center. A monofrequency flow perturbation induces monofrequency flexural vibration of a noncavitating wing, but the vibration of a cavitating wing has multifrequency. Computation of NACA-16009 hydrofoil vibration was made for various free-stream speeds, module of elasticity, fluid, and wing densities, and as a result, three frequency bands of a vibration increase are determined. The low-frequency band is connected with a cavity volume oscillation. There is a considerable influence of cavity length on vibration. The high-frequency band is connected with elastic resonances of the wing. A solitary resonance was found in the middle band. This resonance has a nonlinear dependence on the loss coefficient and does not have a dependence on cavity dimensions and wing elasticity. This resonance appearance is connected with an interaction of lift and media inertia forces.

Interaction of elementary waves for equations of potential flow

Interaction of elementary waves for equations of unsteady potential flow in gas dynamics is considered. Under the assumptions on weakness of strength of the elementary waves the structure of solutions has been given in various cases of interaction of simple wave with shock, or interaction between simple waves or shocks. Hence the complete results on interaction of weak elementary waves for second-order equation of potential flow are obtained.

Thrust generation due to airfoil flapping

Thrust generation on a single flapping airfoil and a flapping/stationary airfoil combination in tandem is studied parametrically. A multiblock Navier-Stokes solver is employed to compute unsteady flowfields. The unsteady flowfield around a single flapping airfoil is also computed by an unsteady potential flow code. The numerical solutions predict thrust generation in flapping airfoils and a significant augmentation of thrust in flapping/stationary airfoil combinations in tandem. The propulsive efficiency is found to be a strong function of reduced frequency and the amplitude of the flapping motion. At a flapping amplitude of 0.40 chord lengths and a reduced frequency of 0.10, the propulsive efficiency of a single NACA 0012 airfoil was computed to be more than 70 %. For the airfoil combination in tandem, the propulsive efficiency was augmented more than 40% at a reduced frequency of 0.75 and a flapping amplitude of 0.20 chord lengths when the airfoils are separated by about two chord lengths.

Nonsteady Stagnation-Point Flows Over Permeable Surfaces: Explicit Solutions of the Nevier-Stokes Equations

Some new explicit solutions of the unsteady two-dimensional Navier-Stokes equations describing nonsteady stagnation-point flows with surface suction or injection are presented. The solutions have been obtained using a new approach for finding explicit similarity solutions of partial differential equations. As distinct from the common Birkhoff’s similarity transformation, which permits only one form of an unsteady potential flow field and only one form of time dependence for suction (or blowing) velocity, the transforms obtained permit consideration of a variety of special solutions differing in the forms of time dependence.

The time-dependent free surface flow induced by a submerged line source or sink

The unsteady nonlinear potential flow induced by a submerged line source or sink is studied by a vortex sheet method, both to trace the free surface evolution and to explore the possible existence of steady-state solutions. Only steady-state flows have been considered by other investigators, and these flows have been insensitive to whether they are generated by a source or sink, except with respect to the flow direction along the streamlines. The time-dependent solution permits an assessment of the stability of previously found steady solutions, and also reveals differences between source and sink flows: for the infinite-depth case, steady stagnation-point-type solutions are found both for source flows and sink flows, above the critical value reported by other investigators; finally, it is shown that streamline patterns of steady stagnation-point flows are identical for source and sink flows only in the limiting case of infinite depth.

Finite element calculations and experimental verification of the unsteady potential flow in a centrifugal volute pump

AbstractIn this paper we present a finite‐element‐based methode for the calculation of the unsteady potential flow in rotor/stator configurations. A numerical algorithm was developed to calculate the two‐dimensional flow through a centrifugal volute pump, taking into account the width variation of the volute in the axial direction and the vortex wakes downstream of the impeller blades by a linearized vortex distribution. The pressure field was obtained from the unsteady Bernoulli equation, with the entire configuration of the pump being included in the calculations. For that purpose the computational domain was split into a region containing the rotor and one containing the stationary parts, each region being treated in a different co‐ordinate system. The corresponding finite element grids qare matched by an interface consisting of connect elements which move with time. The method is applied to a laboratory centrifugal pump set up at the Von Karman Institute for Fluid Dynamics, which made an experimental validation possible. The total head of the pump and the velocity and pressure fields were computed and analysed for various mass flows. The agreement with the experimental data was satisfactory. The deviation was largest at low mass flow, the maximum deviation in the velocity around the impeller being 10 per cent. The overall behaviour of the pump could be well predicted.

Spectral Solutions For Unsteady Annular Flows Between Eccentric Cylinders Induced By Transverse Oscillations

A spectral collocation method is formulated in this paper for the analysis of unsteady annular flows between eccentric cylinders, induced by transverse oscillations of one of the cylinders. This method uses convenient spatial and temporal expansions for the unsteady flow parameters and a collocation approach to determine the a prioriunknown complex coefficients in these spectral expansions. The method is first used to solve the unsteady potential flow between long eccentric cylinders generated by small amplitude translational oscillations. For validation, the resulting solutions for the added mass coefficients are compared with the available ideal flow results for concentric and eccentric configurations. Spectral solutions are then obtained with this method for the more complex problem of unsteady viscous flows. Solutions for the unsteady pressure and fluid-dynamic forces are obtained for oscillations either in the plane of symmetry of the eccentric annular configuration or normal to it; they are compared to existing analytical and finite element viscous flow solutions, for concentric and eccentric cylinders, respectively. A parametric investigation is conducted to expose the influence of the relative eccentricity, outer to inner radius ratio and the oscillatory Reynolds number on the added mass and viscous damping coefficients.

Potential flows of viscous and viscoelastic fluids

Potential flows of incompressible fluids admit a pressure (Bernoulli) equation when the divergence of the stress is a gradient as in inviscid fluids, viscous fluids, linear viscoelastic fluids and second-order fluids. We show that in potential flow without boundary layers the equation balancing drag and acceleration is the same for all these fluids, independent of the viscosity or any viscoelastic parameter, and that the drag is zero when the flow is steady. But, if the potential flow is viewed as an approximation to the actual flow field, the unsteady drag on bubbles in a viscous (and possibly in a viscoelastic) fluid may be approximated by evaluating the dissipation integral of the approximating potential flow because the neglected dissipation in the vorticity layer at the traction-free boundary of the bubble gets smaller as the Reynolds number is increased. Using the potential flow approximation, the actual drag D on a spherical gas bubble of radius a rising with velocity U(t) in a linear viscoelastic liquid of density ρ and shear modules G(s) is estimated to be \[D = \frac{2}{3}\pi a^3 \rho {\dot U} + 12\pi a \int_{-\infty}^t G(t - \tau)U(\tau){\rm d}\tau\] and, in a second-order fluid, \[D = \pi a\left(\frac{2}{3}a^2 \rho + 12\alpha _1\right ) {\dot U} + 12\pi a\mu U,\] where α1, < 0 is the coefficient of the first normal stress and μ is the viscosity of the fluid. Because α1 is negative, we see from this formula that the unsteady normal stresses oppose inertia; that is, oppose the acceleration reaction. When U(t) is slowly varying, the two formulae coincide. For steady flow, we obtain the approximate drag D = 12πaμU for both viscous and viscoelastic fluids. In the case where the dynamic contribution of the interior flow of the bubble cannot be ignored as in the case of liquid bubbles, the dissipation method gives an estimation of the rate of total kinetic energy of the flows instead of the drag. When the dynamic effect of the interior flow is negligible but the density is important, this formula for the rate of total kinetic energy leads to D = (ρa – ρ) VBg · ex – ρaVB U where ρa is the density of the fluid (or air) inside the bubble and VB is the volume of the bubble.Classical theorems of vorticity for potential flow of ideal fluids hold equally for second-order fluid. The drag and lift on two-dimensional bodies of arbitrary cross-section in a potential flow of second-order and linear viscoelastic fluids are the same as in potential flow of an inviscid fluid but the moment M in a linear viscoelastic fluid is given by \[M = M_I + 2 \int_{-\infty}^t [G(t - \tau)\Gamma (\tau)]{\rm d}\tau,\] where MI is the inviscid moment and Γ(t) is the circulation, and \[M = M_I + 2 \mu \Gamma + 2\alpha _1 \partial \Gamma /\partial t\] in a second-order fluid. When Γ(t) is slowly varying, the two formulae for M coincide. For steady flow, they reduce to \[M = M_I + 2 \mu \Gamma ,\] which is also the expression for M in both steady and unsteady potential flow of a viscous fluid. Moreover, when there is no stream, this moment reduces to the actual moment M = 2μΓ on a rotating rod.Potential flows of models of a viscoelastic fluid like Maxwell's are studied. These models do not admit potential flows unless the curl of the divergence of the extra stress vanishes. This leads to an over-determined system of equations for the components of the stress. Special potential flow solutions like uniform flow and simple extension satisfy these extra conditions automatically but other special solutions like the potential vortex can satisfy the equations for some models and not for others.

Forces on bodies moving in non-uniform and unsteady streams

A generalized and simple equation permitting the calculation of the force on a deeply submerged body translating in a non-uniform and unsteady potential flow is presented. This equation is useful when dealing with flows where separation effects are of less importance and, therefore, inertial forces dominate. The developed formulation gives a better insight into the effect of the non-linearities in fluid forces by considering the convective terms in fluid acceleration. It could be considered as an alternative improved formulation in calculating the inertial part of wave forces on small bodies.

Unsteady potential flows computation by cellular automata: The premixed flame instability

Abstract A cellular automaton leading to self-propagating fronts in a discrete fluid has been implemented. In spite of the stabilizing diffusive-convective mechanisms, planar propagation of these fronts becomes unstable due to a feed-back mechanism with the flow generated by the front itself. The stability properties have been analyzed in the continuum limit and compared with the discrete simulations as well as with the stability properties of premixed flame fronts.

Investigation of rotor tip vortex interactions with a body

Experiments were conducted to examine rotor tip vortex interactions with a body in low-speed forward flight. Unsteady pressure measurements were made at points along the top and around the circumference of the body surface. Flow visualization of the rotor wake was performed using the wide-field shadowgraph method. Considerable insight into the tip vortex interaction processes was obtained by correlating the pressure loads with the vortex trajectories as they approached, distorted, and impinged on the body surface. Unsteady potential flow theory was explored as a means of predicting the unsteady pressure loads on the body surface, using prescribed tip vortex trajectories measured from flow visualization. The results have shown that the process of tip vortex interaction with a body can be divided into three regimes: 1) close tip vortex/body interactions, which is an inviscid flow regime; 2) vortex/surface impingement; and 3) postvortex/surface impingement; the latter involves viscous effects. The results have also shown that the pressures at points on the body exhibited a high sensitivity to tip vortex convection speed and location, which makes the general prediction of such interactional phenomena difficult with existing rotor/airframe interaction models.

The Hydroelastic Stability of Three-Dimensional Disturbances of a Finite Compliant Wall

A theoretical investigation of the hydroelastic behaviour of a compliant panel of finite dimensions is presented. The mechanics of the wall are modelled by using a linear differential operator with a single degree of freedom, and the results herein pertain to a spring-backed flexible plate. Unsteady potential flow is assumed; thus, the onset of divergence instability defines the critical flow speed. At higher flow speeds, this static instability may give way to modal coalescence flutter which takes the form of a downstream travelling wave. Comparison is made between the present analysis and an equivalent two-dimensional (infinite width) study for a compliant wall of finite length. It is demonstrated that the simpler two-dimensional methods of instability prediction are always appropriate when the aspect ratio is greater than five. Increasing the streamwise mode number (i.e., reducing the ratio of disturbance wavelength to panel length) renders the system progressively more two-dimensional in its behaviour. The order of the critical mode is promoted by increasing the foundation-spring stiffness, reducing the width of the panel or imposing a higher order transverse mode than the fundamental. For a compliant panel of prescribed surface area, a critical aspect ratio exists in which configuration the lowest divergence-onset flow speed is found. Arrays of compliant panels are considered: the influence of neighbouring panels exercises a mildly adverse effect on the stability of a panel in a comparison with the response of a similar panel in isolation.

Boundary element analysis of unsteady aerodynamics of windmill rotors in the presence of yaw

In this paper a method for the evaluation of pressure and velocity fields for a horizontal-axis wind turbine in the presence of yaw is presented. The flow is assumed to be potential and incompressible. The formulation used is general and is applicable to unsteady incompressible potential flows around wind turbines, as well as aircraft and ship propellers. In order to assess the importance of the wake geometry a free-wake analysis is included in this paper. Comparisons with experimental results are also included.

Unsteady potential and viscous flows between eccentric cylinders

This paper presents a newly developed spectral collocation method for the study of the unsteady annular flow between two eccentric cylinders. In order to predict the stability of a system in a confined flow, the formulae and results of added mass and fluid damping are provided in the present paper when a cylinder undergoes oscillatory motion in the plane of symmetry and normal to the plane of symmetry in an eccentric annulus. The potential flow theory has been developed to obtain the added mass for incompressible, inviscid and irrotational fluid. For the viscous fluid, the added mass and the viscous damping are presented. This method is validated by comparison with the available analytical solutions obtained for the unsteady potential flow in the eccentric annular space. Excellent agreement was found between the solutions obtained with the present spectral method and the available analytical solutions. In the present study, the viscous effect on the added mass can be evaluated, comparing the results obtained by potential flow theory with those obtained by the viscous flow theory, and viscous damping is investigated.

The Effect of Steady Aerodynamic Loading on the Flutter Stability of Turbomachinery Blading

An aeroelastic analysis is presented that accounts for the effect of steady aerodynamic loading on the aeroelastic stability of a cascade of compressor blades. The aeroelastic model is a two-degree-of-freedom model having bending and torsional displacements. A linearized unsteady potential flow theory is used to determine the unsteady aerodynamic response coefficients for the aeroelastic analysis. The steady aerodynamic loading was caused by the addition of (1) airfoil thickness and camber and (2) steady flow incidence. The importance of steady loading on the airfoil unsteady pressure distribution is demonstrated. Additionally, the effect of the steady loading on the tuned flutter behavior and flutter boundaries indicates that neglecting either airfoil thickness, camber, or incidence could result in nonconservative estimates of flutter behavior.

The asymptotic solution of the unsteady planar parallel gas flow for the case that the adiabatic index number γ is big

In ref. l. under the condition that the components of velocity are only the functions of time and polar angle 0. Dvornikov solved eqs. (1.1)–(1.3) of the ideal gas unsteady planar parallel potential flow. It was pointed out in ref.[1] that in general cases, the evident solutions could not be obtained. Only for two especial cases, the evident solutions were obtained. In this paper, the author studies the same problem as that in ref. [1]. In the first section we obtain the evident solution of equations (1.1)–(1.3) under the condition that the sonic velocity is restricted by some complemental conditions. In the second section, we obtain the first-order approximate solutions of the fundamental equation for the case that

A STUDY OF THE HYDROELASTIC STABILITY OF A COMPLIANT PANEL USING NUMERICAL METHODS

A numerical method is developed which can simulate the interaction between a finite compliant panel and an unsteady potential flow. A boundary‐element technique yields the flow solution whilst finite‐differences are used to solve the wall dynamics; these are then coupled to generate a fully interactive wall/flow system. Thus, the evolution of any wall disturbance can be followed. Parallel computing is appropriately employed and a stability investigation of a realistic compliant panel is carried out. Three‐dimensional flexural waves are found below a critical flow speed whilst beyond this threshold, essentially two‐dimensional unstable divergence waves are found. The form of divergence shows good agreement with that seen in experimental studies. The versatility of this new method will permit the investigation of a wide variety of single‐ and multi‐panel configurations subject to different forms of excitation.

A numerical simulation of the interaction of a compliant wall and inviscid flow

A method for numerically simulating the hydroelastic behaviour of a passive compliant wall of finite dimensions is presented. Using unsteady potential flow, the perturbation pressures which arise from wall disturbances of arbitrary form are calculated through a specially developed boundary-element method. These pressures may then be coupled to a suitable solution procedure for the wall mechanics to produce an interactive model for the wall/flow system. The method is used to study the two-dimensional disturbances which may occur on a Kramer-type compliant wall of finite length. Finite-difference methods are used to yield wall solutions driven by the fluid pressure after some perturbation from the equilibrium position. Thus, histories of surface deflection and wall energy are obtained. Such a modelling of the physics of the system requires no presupposition of disturbance form.A thorough investigation of divergence instability is carried out. Most of the results presented in this paper concern the response of the compliant wall while (and after) a point pressure pulse, carried in the applied flow, travels over the compliant panel. Above a critical flow speed and once sufficient time has passed, the compliant wall is shown to adopt the particular profile of an unstable mode. After this divergence mode has been established, instability is realized as a slowly travelling downstream wave. These features are in agreement with the findings of experimental studies. The role of wall damping is clarified: damping serves only to reduce the growth rate of the instability, leaving its onset flow speed unchanged. The present predictions provide an improvement upon some of the unrealistic aspects of predictions yielded by travelling-wave and standing-wave treatments of divergence instability.The response of a long compliant panel after a single-point pressure-pulse initiation, applied at its midpoint, is simulated. At flow speeds higher than a critical value, parts of the formerly (at subcritical flow speeds) upstream-travelling wave system change to travel downstream and show amplitude growth. The development of this ‘upstream-incoming’ wave illustrates how divergence instability can occur at locations upstream of the point of initial excitation. Faster flexural waves transmit energy upstream, thereafter these disturbances can evolve into slow downstreamtravelling divergence waves. The spread of the instability to locations both downstream and upstream of the point of initial excitation indicates that divergence is an absolute instability. This behaviour and the effects of wall damping clarified by the present work strongly suggest that divergence is a Class C instability.