Chapter 15 - Compressible Flow

Compressible flow past thin wings and slender bodies are studied. First, subsonic flow past thin wings are analyzed by means of potential flow theory. The kernel function method is introduced for arbitrary planforms undergoing simple harmonic oscillations. The spanwise polynomial approximation and the chordwise trigonometric function approximations which automatically satisfy the Kutta condition and inherit the leading edge singularity are used. The Doublet-Lattice method as a more general numerical approach is given for the analysis of the flow past additional surfaces like tail or store surfaces, which are not necessarily in plane with the wing surface or surfaces having spanwise deflections. Airfoil response to the arbitrary unsteady motion is also given for subsonic flows. A brief review of shock waves and Mach waves in a supersonic flow is given. Afterwards, unsteady supersonic potential flow is studied for a simple harmonically oscillating point source. First, unsteady flow past an airfoil is considered. Then, supersonic flow about thin wings are analyzed using the Mach box technique. Introduction to supersonic kernel function method is briefly provided. Arbitrary unsteady motion of an airfoil in a supersonic flow is presented. Slender body theory is introduced to analyze the cross flow past bodies of considerable fineness where the cross flow is shown to be approximately incompressible under certain conditions. In connection with the slender body approximation, the Munk’s airship theory is utilized to predict the stability derivatives of missile like bodies.

We have modelled ultracompact HII regions (UCHIIR) in terms of steady subsonic ionized flows in a clumpy medium. Mass loss from neutral clumps allows the regions to be long-lived. We examine the form of global flows for different dependences of the volume mass injection rate, $\dot{q}$, on radius and Mach number, and describe the solutions in detail. We find that three observed UCHIIR morphologies are reproduced with these models. Mach number independent flows that include a radial variation can give centre- brightened core--halo morphologies. Mach number dependent flows reproduce naturally the uniform UCHIIR morphology. In a hybrid model, including subsonic and supersonic flows, we allow a supersonic wind to shock in the ionized region. The ionized subsonic gas has a high density and so dominates the emission. The shell produced has a velocity structure very different from that of fully supersonic models. Several morphologies of spherical UCHIIR can be understood in terms of these various models; however, kinematic data are crucial as a discriminant between them.

An earlier analysis of a viscous mixing region with heat addition interacting with a supersonic external flow has been extended to include cases with a sloped wall beneath the viscous region. Also, cases with a supersonic to subsonic transition at the viscous throat have been studied for the first time. Limitations on the allowable heating value of the fuel that correspond to smooth subsonic to supersonic, or supersonic to subsonic transitions, are delineated and explained. The effects of wall slope at the throat and external Mach number on these limitations are shown. The results of calculations for representative cases show that it is possible to cancel base drag and even produce net thrust for cases with downward sloped walls if the initial Mach number in the viscous zone is supersonic prior to addition of heat. This corresponds to a supersonic to subsonic transition. For an initially subsonic flow, the results indicate pressures below the freestream static pressure throughout the flowfield. This means that only very limited drag from flameholders, such that the flow remains supersonic before heat addition, are advisable. Finally, some directions for future work are outlined.

The nozzle is a kind of equipment with a wide range of usage in the aerospace industry, of which its performance is vital for rocket engines by changing the geometry of the inner wall of the pipe section to accelerate the airflow. There are two types of nozzles commonly used: one is a tapered nozzle, and the other is a Laval nozzle. To analyze the flow phenomenon in nozzle is vital before transferring prototype design into industrial production, whether by laboratory experiment or computational simulation. In this paper, two different numerical methods are adopted to simulate gas behavior inside a simplified nozzle with upper and bottom symmetrical bumps. The first is to solve one dimensional Euler equations, and the other is to solve a scalar variable named velocity potential with small disturbance equations (SDE). The solutions under various inlet Mach numbers are compared by analysing the velocity fields and Mach number contours obtained by these two approaches. Similarities and differences between the Euler method and the potential SDE method for subsonic flow and supersonic flow are the key emphases in this work. For either subsonic or supersonic flow, the Mach number distribution along the nozzle’s center line shows a consistent trend for both methods. In contrast, the values of maximum or minimum Mach number have corresponding differences. Moreover, by using potential SDE simulation, several types of shock waves are successfully captured. All results show that the incoming airflow decelerates at the leading edge of the nozzle then accelerates when passing through the bumps, and finally decelerates back to the speed of inlet flow. The difference is that flows with varied Mach numbers has distinct velocity distributions in the nozzle.

Compressible-gas flow through convergent and convergent-divergent nozzles is analysed in this chapter based upon the conservation laws for mass, momentum, and energy, together with considerations of thermodynamics. It is shown that in both cases the key parameter in describing the flow is the Mach number, which is used to distinguish between subsonic and supersonic flow. So that significant results can be achieved, the flowing fluid is treated as a perfect gas, and the flow as one dimensional. Flow through a convergent nozzle and the choking limitation is discussed. Flow through a normal shockwave, which is an important feature of supersonic flow, is also analysed. No account is taken of surface friction or heat transfer, and the flow upstream and downstream of a shockwave is treated as isentropic. In addition, the conditions are discussed under which a shockwave arises in compressible flow through a convergent-divergent nozzle.

In this article, a comparison is made between the robustness and efficiency of the CLEAR algorithm and the SIMPLE algorithm on nonorthogonal curvilinear coordinates for compressible flows. Thirteen different high-order convection schemes are employed in the calculations. Subsonic flow, transsonic flow, and supersonic flow in a channel with a circular arc bump and compressible flow in a Laval nozzle are used as test cases. The CLEAR algorithm shows huge potential to compute the transsonic flow in the Laval nozzle and high-speed compressible flows. Results with the ADBQUICKEST scheme, the HLPA scheme, and the MUSCL scheme are stable for both the compressible SIMPLE and CLEAR algorithms for all the mentioned cases.

In order to find some effects of compressibility on the flow of a conducting gas past a magnetized body, the two-dimensional, steady flow of a weakly conducting gas past a plane wall containing a magnetic dipole, the axis of which is placed parallel to the wall, is considered on the basis of the small perturbation theory. The boundary value problem in terms of perturbation velocity is formulated and solved for subsonic and supersonic flows. In subsonic flow of a comparatively small Mach number, the drag and moment depend only on a parameter N , where N = R m S , R m is the magnetic Reynolds number and S is the pressure number. In supersonic flow, the velocity distribution on the wall is not monotonus, and becomes more irregular as the Mach number approaches unity.

In the initial article 1 of this series, we described our methods of investigating intracranial hydrodynamics in human cadavers under experimental conditions that rendered certain of our results applicable to the living. Two of our conclusions were: The lateral ventricles and the cisternal and lumbar subarachnoid spaces remained in free communication under all experimental variations in pressure, provided no distortion of the subtentorial structures was artificially produced. In the absence of such distortion there was no evidence of tentorial, medullary or cerebellar movement under any experimental variations in subarachnoid fluid pressure. Specifically, we could not produce descent of the cerebellum or medulla into the foramen magnum by hydrodynamic means alone, e.g., sudden release of high intracranial fluid pressure through rapid lumbar or cisternal tap. We were therefore unable to confirm the generally accepted importance of the mechanical factor in the etiology of cerebellar herniation and bulbar compression in cases in

The results of some conceptual studies on the development and aerodynamic design of lifting configurations feasible for hypersonic vehicles powered by airbreathing engines are presented and discussed. The configurations, including the waveriders, with three-dimensional (3D), so-called convergent inlets, were considered. These inlets are designed so that the supersonic flow over them has the streamlines converging in space. Besides, they have a compact throat cross-section shape close to a circle. The latter property facilitates the engine heat protection as against the conventional 2D (flat) inlets or axi-symmetric inlets with the centerbodv having the slotted throats. The research included, firstly, a class of convergent inlets and lifting configurations designed by combining the regions of supersonic inviscid uniform flows formed over swept wedges and V-shaped bodies. Another class of configurations is designed with the use of supersonic axisymmetric inviscid compression flows. Swept concave surfaces contoured to form 3D supersonic convergent compression flows were also studied. Under the said aerodynamic design a wide set of numerical methods from the simplest quadratic ones to the 2D and 3D ones was used. The results of parametric calculations of the configurations built are considered and, in particular, some questions concerning the integration of convergent inlets with the lifting surfaces are discussed.

This paper presents simple and efficient analytical solutions for unsteady subsonic compressible flows past flexible airfoils executing low frequency oscillations. These analytical solutions are obtained with a method using velocity singularities related to the airfoil leading edge and ridges. The method has been validated for the pitching and plunging oscillations of the rigid airfoils by comparison with results based on Jordan’s data for compressible flows and by comparison with the solutions obtained by Theodorsen, Postel & Leppert and Mateescu & Abdo for incompressible flows. The method has been applied to obtain efficient analytical solutions for the flexural oscillations of airfoils in compressible flows, which can be efficiently used in solving aeroelastic problems. Professor Carafoli’s contributions to the aerodynamics of airfoils and wings Starting from 1924, in Paris, Elie Carafoli made significant contributions to the aerodynamic studies of wings and airfoils, included in Aerodynamics of airplane wings [5], Theory of lifting airfoils [6], in which he presented the aerodynamics of airfoils with rounded trailing edge, named later Carafoli airfoils, Influence of ailerons on the aerodynamic characteristics of lifting surfaces [7], and in Experimental studies on monoplane wings [8]. After 1928, he continued his research and academic activity in Romania, bringing new contributions to the aerodynamics of wings and airfoils in incompressible flows, which were mostly included in Aerodynamics [9], book also translated from Romanian into German and Russian. After 1950, Elie Carafoli developed a special interest for compressible flows and published High Speed Aerodynamics [10], initially in Romanian and then translated in English. He made, together with his collaborators, important contributions on the aerodynamics of wings and airfoils in supersonic flows, most of them included in the monograph Wing Theory in Supersonic Flow [11] published by Pergamon Press, to which the present author also collaborated. INCAS BULLETIN No. 2/ 2009 51 DOI: 10.13111/2066-8201.2009.1.2.11 For his outstanding achievements in aerodynamics, Elie Carafoli received very numerous international distinctions and awards, such as Diploma “Paul Tisandier” of the International Aeronautical Federation, Gauss Medal attributed yearly to the best scientist in the world by the Scientific Society Braunschweig, Germany, Apollo 11 Medal, Tsiolkovski Medal and many others distinctions which can not be listed here due to the space limitation. Elie Carafoli was one of the few Honorary Fellows of the Royal Aeronautical Society, Great Britain, and one of the first members of the International Academy of Astronautics (proposed by Theodor von Karman), and he was elected first Vice-President and then President of the International Astronautic Federation. In Romania, Carafoli was elected member of the Romanian Academy in 1948, and later named President of the Technical Sciences Section of the Academy, and he received the greatest Romanian medals, prizes and awards. One of the important realizations of Academician Carafoli was the foundation in 1949 of a research institute of the Romanian Academy, known as the Institute of Fluid Mechanics, which is presently named the National Institute for Aerospace Research “Elie Carafoli”. This institute contributed to the scientific formation of numerous former students and collaborators of Professor Carafoli, who distinguished themselves by contributions in aerodynamics and other related aerospace domains. Many of the former students and collaborators of Professor Carafoli became professors at their turn in Romania, as well as in other countries, such as Canada, France, Germany and United States of America, to name only a few. The present author had the privilege to study Aerodynamics with Professor Carafoli, who was also the Ph. D. supervisor, and to be his close collaborator for more than two decades. He is indebted to Professor Carafoli for many research advices, which materialized not only in many publications resulted from their direct scientific collaboration, but also in his further research and academic activity, first in Romania, and since 1982 in Canada. This author, together with his Masters’ and Ph. D. students, aims to continue the best scientific tradition of Professor Carafoli, with new contributions to the aerodynamics of wings and airfoils in steady and unsteady flows in incompressible, compressible subsonic and supersonic regimes and at low Reynolds numbers, in addition to other contributions in related aerospace domains (a few selected examples from the last ten years are included in references [20–37, 43].

Tracer pathline statistics have rarely, if at all, been used in the investigation of compressible subsonic, transonic and supersonic flows past a circular cylinder. Here, we report pathline histogram, conditional histogram and zonal histogram data, focusing on the characteristics of particle instantaneous flow reversal and their dependencies on Mach number and tracer release position. The far-upstream Mach number and the diameter-based Reynolds number used for these three flow regimes are ( 0.2 , 10000 ) , ( 0.9 , 3900 ) and ( 1.2 , 10000 ) , respectively. The subsonic surface-release histogram profiles display unexpected persistent oscillations at a frequency 50 times of the periodic vortex shedding frequency, reflecting the effect of a very small magnitude zig-zagging type of particle motion in a region very close to the wall and slightly behind the mean flow separation location. No similar high-frequency oscillation is observed in the corresponding transonic and supersonic histograms likely because, at Ma ≥ 0.9 , periodic vortex shedding ceases to exist and the near-wake becomes a quasi-laminar recirculation zone bounded by a pair of converging slip-layers with a neck opening to the far-wake. At the instant of 5 subsonic vortex-shedding periods after being released from the cylinder surface, there are 1.67 % , 35.2 % and 39.7 % particles that experience flow reversal in the subsonic, transonic and supersonic flows, respectively. Surprisingly, in the present transonic flow, growth of the turbulent far-wake behind the pair of lambda-shocks is constrained over a considerable streamwise distance.

The aerodynamic efficiency of a rocket is significantly influenced by its fin design. This study explored the use of bio-inspired fin geometries on shark skin riblets and humpback whale tubercles to reduce drag on a model rocket in both subsonic (30 m/s) and supersonic (Mach 1) flow regimes using Computational Fluid Dynamics (CFD). Simulations in ANSYS Fluent were performed for a conventional smooth fin (baseline) and for fins with riblet and tubercle modifications. The bio-inspired designs altered the boundary layer on the fin surfaces, affecting flow separation and pressure distribution. Key results indicated that whale-inspired tubercle fins consistently lowered drag in both speed regimes by up to 11%, by delaying flow separation and reducing pressure drag. Shark skin riblets displayed mixed performance: they increased drag at low speed, due to higher skin friction, but achieved a 6% drag reduction at Mach 1 by mitigating form drag in the wake. These findings demonstrate the potential of nature’s geometries on rocket fins to strategically leverage turbulence for drag reduction.

A correspondence between the solutions of the direct and the inverse problem for wing theory is established for a wing of finite span in the framework of linear theory on the basis of solution of a wave equation in Volterra form for supersonic flow and solution of the Laplace equation in the form of Green's formula for subsonic flow. For the direct problem in the case of supersonic flow an expression is derived for finding the load on the wing with maximal allowance for the wing geometry. In the inverse problem for supersonic and subsonic flows, expressions are derived for finding the wing geometry from given values of the load on the wing and the variation of the load along the span of the wing. The solution of the inverse problem is presented in the form of integrals that converge for interior points of the wing surface in the sense of the Cauchy principal value, the wing surface being represented as a vortex surface of mutually orthogonal vortex lines. The conditions of finiteness of the velocities on the edges are discussed.

The Transonic Small Disturbance (TSD) Equation is a common model equation for describing subsonic and supersonic flow close to the local speed of sound (transonic). In transonic flow there is an embedded region of locally supersonic flow inside an otherwise subsonic flow. The supersonic region is usually terminated by a shock discontinuity. The success of a numerical scheme for transonic flow prediction depends on its capability of capturing all the flow details and non-linearities including sharp shock profiles without oscillations near the shock. Most of the important phenomena in the TSD equation occur in the stream-wise direction. The nonlinearity and changes in the region of influence depend only on the stream-wise derivation. A suitable one-dimensional model equation derived from the TSD Equation is used. The one-dimensional equation is written in conservation law form. This nonlinear system of equations models the transition from supersonic to subsonic flow. In the numerical calculations the discretised problem is treated as a series of Riemann problems. We will investigate various techniques for solving these Riemann problems. It will be shown that the techniques do not allow non-physical expansion shocks to develop and that the techniques smooth out expansion shocks when these non-physical shocks are present in the initial velocity profile. A comparison will be made between the schemes based on the sharpness of the resulting shock profiles.

This study presents flow simulation over the reentry capsule at supersonic and hypersonic speeds. Numerical algorithms solve for the CFD method, which is produced using help ANSYS Fluent 19.2. The using GPU core to get a solution faster. The main purpose – flow simulation and numerical analysis reentry capsule; understand the behavior of supersonic and hypersonic flow and its effect on the reentry capsule; compare temperature results for the range Mach numbers equals 2–6. This study showed results on velocity counters, on temperature counters and vector of velocity for range Mach numbers equals 2–6. This study demonstrates the importance of understanding the effects of shock waves and illustrates how the shock wave changes as the Mach number increases. For every solves, the mesh had adapted for pressure gradient and velocity gradient to get the exact solution. As a result of the obtained solution, it is found that a curved shock wave appears in front of the reentry capsule. The central part of which is a forward shock. An angular expansion process is observed, which is a modified picture of the Prandtl- Mayer flow that occurs in a supersonic flow near the sharp edge of the expanding region. It is revealed that with an increase in the Mach number, the shock wave approaches the bottom of the reentry capsule, and there is also a slope of the shock to the flow direction, with an increase in the Mach number. The relevance and significance of this problem for the design of new and modernization of old reentry capsules.