Compressible Potential Flow Research Articles

Convexity of shock polars

Abstract We show that the shock polars of compressible full potential flow are strictly convex if the enthalpy per mass is a convex function of volume per mass, in particular when the sound speed is a non-decreasing function of density. Counterexamples are given for some cases that violate the condition on enthalpy. For the full Euler equations with convex equation of state satisfying the ideal gas law, polars are strictly convex if heat capacity is constant, but counterexamples are given in variable cases, showing no useful generalizations are possible.

Meshless Fragile Points Method (FPM) in a 2D and 3D potential compressible subsonic fluid flow

The possibility of the implementation of Fragile Points Method (FPM) in the analysis and solving of the fluid flow problems is demonstrated in this paper. Subject of interest was the compressible potential flow equation, which was discretised by using the meshless Galerkin FPM. Discontinuous and point-based trial and test functions were used, with the introduction of Numerical Flux Corrections for overcoming the inaccuracy and instability side-effects brought by discontinuities. By comparison with the analytical, experimental and numerical results available in literature, FPM was verified, and it was shown that FPM could be very good tool for the potential compressible fluid flow simulations.

Shock polars for non-polytropic compressible potential flow

<p style='text-indent:20px;'>We consider compressible potential flow for general equations of state. Assuming hyperbolicity and convex equation of state, we prove that shock polars have a unique critical point (in each half), as well as a unique sonic point, with critical and strong shocks always on the subsonic side. We also show existence of normal and oblique shocks, as well as monotonicity of density, enthalpy, pressure along each half-polar, with Mach number monotone only along the subsonic part.</p>

Compressible potential flows around round bodies: Janzen–Rayleigh expansion inferences

The subsonic, compressible, potential flow around a hypersphere can be derived using the Janzen–Rayleigh expansion (JRE) of the flow potential in even powers of the incident Mach number ${\mathcal {M}}_\infty$ . JREs were carried out with terms polynomial in the inverse radius $r^{-1}$ to high orders in two dimensions, but were limited to order ${\mathcal {M}}_\infty ^{4}$ in three dimensions. We derive general JRE formulae for arbitrary order, adiabatic index and dimension. We find that powers of $\ln (r)$ can creep into the expansion, and are essential in the three-dimensional (3-D) sphere beyond order ${\mathcal {M}}_\infty ^{4}$ . Such terms are apparently absent in the 2-D disk, as we verify up to order ${\mathcal {M}}_\infty ^{100}$ , although they do appear in other dimensions (e.g. at order ${\mathcal {M}}_\infty ^{2}$ in four dimensions). An exploration of various 2-D and 3-D bodies suggests a topological connection, with logarithmic terms emerging when the flow is simply connected. Our results have additional physical implications. They are used to improve the hodograph-based approximation for the flow in front of a sphere. The symmetry-axis velocity profiles of axisymmetric flows around different prolate spheroids are approximately related to each other by a simple, Mach-independent scaling.

Bandgap characteristics of phononic crystals in steady and unsteady flows.

Over the past 30 years, most phononic crystal research has been done for a stationary medium. As reported in a recent experimental study, phononic crystals cannot preserve their bandgaps in the presence of flow. In this study, the bandgap characteristics of a two-dimensional phononic crystal in steady and unsteady flows are investigated theoretically. To identify the effect of the flow on sound insulation in the bandgap frequency ranges, the acoustic reflectance spectra of phononic crystals for different types of background flows, including a uniform flow, a compressible potential flow, and a turbulent flow were calculated. For the steady flows, which include uniform and compressible potential flows, the reflectance spectra are shifted to a lower frequency by the factor 1-M2 due to convection when the flow is in the same direction as the incident wave. Moreover, the reflectance spectra of a phononic crystal in a turbulent flow were evaluated for various combinations of inflow speeds and geometric parameters, such as the filling ratio and the number of layers. Due to the aerodynamic noise and fluid convection, a phononic crystal cannot work as an acoustic barrier, rather it becomes an aeroacoustic source in a turbulent flow.

AN INVERSE DESIGN METHOD FOR A WING IN FULL CRUISING CONFIGURATION OF A REGIONAL AIRCRAFT VIA RANS APPROACH

The previously developed iteration method of the wing geometry construction by the given pressure distribution in the compressible viscous flow is applied to the wing in the full configuration of the subsonic aircraft including pylons and engine nacelles. The algorithm of the residual correction is used in which the direct calculation of the configuration in the frames of the averaged Navier-Stokes equations is carried out turn by turn with the geometry correction which reduces the discrepancy between the current and the target pressure distribution. As a correction block, the previously developed method for solving the inverse problem for a wing in the same configuration (but without a pylon) in a compressible gas potential flow is used, taking into account the influence of the thickness of the displacement of the boundary layer and the wake of the wing. Examples of wing geometry design with given pressure distribution showing high speed of convergence (three to eight iterations) are presented.

Compressible Pipe Flow with Friction and Gravity

A viscous one-dimensional compressible pipe flow under gravity effect is studied analytically. The compressible one-dimensional pipe flow with friction is called Fanno flow and the solution is given by analytical formula. In gas dynamics, the gravity effect is minimal and it is not included in the equations. However, it was shown by the present author that the elevation of a pipe could change the flow conditions in a one-dimensional compressible potential flow under gravity. The sonic condition is reached at the maximum height for an inviscid pipe flow. In this paper, the gravity effect is extended to the viscous one- dimensional pipe flow. Subsonic–supersonic transition is also possible by up and down of the pipe as in the inviscid flow, and it is found that the sonic condition deviates from the peak position of the pipe.

A Variational Inequality Formulation for Transonic Compressible Steady Potential Flows: Radially Symmetric Transonic Shock

We establish a variational inequality formulation that captures the transonic shock for a steady compressible potential flow. Its critical point satisfies the transonic equation; moreover the associated jump conditions across its free boundary match the usual Rankine--Hugoniot jump conditions for a shock. By means of example we validate our formulation and establish the necessary and sufficient condition for the existence of a transonic shock. Numerical results are also discussed.

Relative entropy and compressible potential flow

Compressible (full) potential flow is expressed as an equivalent first-order system of conservation laws for density ▪ and velocity v. Energy E is shown to be the only nontrivial entropy for that system in multiple space dimensions, and it is strictly convex in ρ,v if and only if |v|<c. For motivation some simple variations on the relative entropy theme of Dafermos/DiPerna are given, for example that smooth regions of weak entropy solutions shrink at finite speed, and that smooth solutions force solutions of singular entropy-compatible perturbations to converge to them. We conjecture that entropy weak solutions of compressible potential flow are unique, in contrast to the known counterexamples for the Euler equations.

Perturbation method for the second-order nonlinear effect of focused acoustic field around a scatterer in an ideal fluid

Two nonlinear models are proposed to investigate the focused acoustic waves that the nonlinear effects will be important inside the liquid around the scatterer. Firstly, the one dimensional solutions for the widely used Westervelt equation with different coordinates are obtained based on the perturbation method with the second order nonlinear terms. Then, by introducing the small parameter (Mach number), a dimensionless formulation and asymptotic perturbation expansion via the compressible potential flow theory is applied. This model permits the decoupling between the velocity potential and enthalpy to second order, with the first potential solutions satisfying the linear wave equation (Helmholtz equation), whereas the second order solutions are associated with the linear non-homogeneous equation. Based on the model, the local nonlinear effects of focused acoustic waves on certain volume are studied in which the findings may have important implications for bubble cavitation/initiation via focused ultrasound called HIFU (High Intensity Focused Ultrasound). The calculated results show that for the domain encompassing less than ten times the radius away from the center of the scatterer, the non-linear effect exerts a significant influence on the focused high intensity acoustic wave. Moreover, at the comparatively higher frequencies, for the model of spherical wave, a lower Mach number may result in stronger nonlinear effects.

Compressible subsonic potential flow past a 2D given sharp angular unbounded domain

In this paper, we focus on the two-dimensional subsonic flow problem around an infinite long ramp. The flow is assumed to be steady, isentropic and irrotational, namely, the movement of the flow is described by a second elliptic equation. By the use of a separation variable method, Strum-Liouville theorem and scaling technique, we show that a nontrivial subsonic flow around the infinite long ramp does not exist under some certain assumptions on the potential flow with a low Mach number.

Analysis of compressible potential flow over aerofoils using the dual reciprocity method

AbstractThe use of the linearised potential model for the analysis of compressible flows is quite widespread, and provides good results for subsonic and supersonic flows. However, the calculation of aerofoils and wings subject to transonic flows requires a non-linear model, such as the transonic small-disturbance (TSD) potential equation. The solution of the problem by a singularity distribution requires singularities over the field, as well as panels on the boundary, characterising the procedure known as field panel method. The present work shows results of calculations of the transonic small-disturbance potential equation for flows without shock waves using the dual reciprocity method (DRM), which permits calculation of integrals only at the boundary of the problem, without the need of field distributions. This approach, compared to the field panel methods, takes considerably less computer time, and shows a significant improvement when compared to results of linear theory without much additional computer time, making this technique adequate to design phases of aircraft. Pressure distribution results show good agreement with other methods found in litterature. The low computational cost of the present method allows us to perform parametric tests and explore the effects of thickness and Mach number on the lift and pitching moment coefficients. A discussion of the physical effect of these parameters on the problem is presented, and the thickness of the aerofoil is shown to increase the lift and change the position of the aerodynamic centre. However, this non-linear effect depends on the precise shape of the thickness distribution.

Non-existence of strong regular reflections in self-similar potential flow

We consider shock reflection which has a well-known local non-uniqueness: the reflected shock can be either of two choices, called weak and strong. We consider cases where existence of a global solution with weak reflected shock has been proven, for compressible potential flow. If there was a global strong-shock solution as well, then potential flow would be ill-posed. However, we prove non-existence of strong-shock analogues in a natural class of candidates.

Transonic Shocks in Multidimensional Divergent Nozzles

We establish existence, uniqueness and stability of transonic shocks for steady compressible non-isentropic potential flow system in a multidimensional divergent nozzle with an arbitrary smooth cross-section, for a prescribed exit pressure. The proof is based on solving a free boundary problem for a system of partial differential equations consisting of an elliptic equation and a transport equation. In the process, we obtain unique solvability for a class of transport equations with velocity fields of weak regularity(non-Lipschitz), an infinite dimensional weak implicit mapping theorem which does not require continuous Frechet differentiability, and regularity theory for a class of elliptic partial differential equations with discontinuous oblique boundary conditions.

Non-spherical bubble dynamics in a compressible liquid. Part 2. Acoustic standing wave

This paper investigates the behaviour of a non-spherical cavitation bubble in an acoustic standing wave. The study has important applications to sonochemistry and in understanding features of therapeutic ultrasound in the megahertz range, extending our understanding of bubble behaviour in the highly nonlinear regime where jet and toroidal bubble formation may be important. The theory developed herein represents a further development of the material presented in Part 1 of this paper (Wang &amp; Blake,J. Fluid Mech. vol. 659, 2010, pp. 191–224) to a standing wave, including repeated topological changes from a singly to a multiply connected bubble. The fluid mechanics is assumed to be compressible potential flow. Matched asymptotic expansions for an inner and outer flow are performed to second order in terms of a small parameter, the bubble-wall Mach number, leading to weakly compressible flow formulation of the problem. The method allows the development of a computational model for non-spherical bubbles by using a modified boundary-integral method. The computations show that the bubble remains approximately of a spherical shape when the acoustic pressure is small or is initiated at the node or antinode of the acoustic pressure field. When initiated between the node and antinode at higher acoustic pressures, the bubble loses its spherical shape at the end of the collapse phase after only a few oscillations. A high-speed liquid bubble jet forms and is directed towards the node, impacting the opposite bubble surface and penetrating through the bubble to form a toroidal bubble. The bubble first rebounds in a toroidal form but re-combines to a singly connected bubble, expanding continuously and gradually returning to a near spherical shape. These processes are repeated in the next oscillation.

Uniqueness and instability of subsonic–sonic potential flow in a convergent approximate nozzle

We proved uniqueness and instability of the symmetric subsonic— sonic flow solution of the compressible potential flow equation in a surface with convergent areas of cross―sections. Such a surface may be regarded as an approximation of a two―dimensional convergent nozzle in aerodynamics. Mathematically these are uniqueness and nonexistence results of a nonlinear degenerate elliptic equation with Bernoulli type boundary conditions. The proof depends on maximum principles and a generalized Hopf boundary point lemma which was proved in the paper.

Unique subsonic compressible potential flows in three -dimensional ducts

We establish the uniqueness of subsonic potential flows in athree-dimensional finite duct, a semi-infinite duct and aninfinite duct with quadrate sections, as well as flows in half spaceand whole space. Moreover, some extremum principles for the relatedelliptic equations are proved under suitable assumptions inunbounded domains.

Global subsonic compressible flows through a two-dimensional periodic duct

We deal with two-dimensional compressible potential subsonic flows in an infinitely long duct with periodic walls. It is shown that there exists a critical value of mass flux: If the incoming mass flux is less than the critical value, then the flow is also periodic. Existence, uniqueness and regularity of the periodic solution are obtained by techniques of elliptic equations.

Global Uniqueness of Transonic Shocks in Divergent Nozzles for Steady Potential Flows

We show that for steady compressible potential flow in a class of straight divergent nozzles with arbitrary cross-section, if the flow is supersonic and spherically symmetric at the entry and the given pressure (resp., velocity) is appropriately large (resp., small) and also spherically symmetric at the exit, then there exists uniquely one transonic shock in the nozzle. In addition, the shock-front and the supersonic flow ahead of it, as well as the subsonic flow behind it, are all spherically symmetric. This is a global uniqueness result of free boundary problems of elliptic-hyperbolic mixed type equations. The proof depends on the maximum principles and judicious choices of comparison functions.

Adaptive least‐squares finite element approximations to transonic‐flow problems

AbstractThis work concerns solutions of compressible potential flow problems based on weighted least‐squares finite element approximations. The model problem considered is that of the potential flow past a circular cylinder. To capture the transonic flow region, an adaptive algorithm based on mesh redistribution with local mesh refinement and smoothing is developed for suitably weighted least‐squares approximations. Numerical results for the model problem are given for the transonic case with shocks. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)