Subsonic and Supersonic Flows

Slender body theory facilitates computational simulations of thin fibers immersed in a viscous fluid by approximating each fiber using only the geometry of the fiber centerline curve and the line force density along it. However, it has been unclear how well slender body theory actually approximates Stokes flow about a thin but truly three‐dimensional fiber, in part due to the fact that simply prescribing data along a 1D curve does not result in a well‐posed boundary value problem for the Stokes equations in ℝ3. Here, we introduce a PDE problem to which slender body theory (SBT) provides an approximation, thereby placing SBT on firm theoretical footing. The slender body PDE is a new type of boundary value problem for Stokes flow where partial Dirichlet and partial Neumann conditions are specified everywhere along the fiber surface. Given only a 1D force density along a closed fiber, we show that the flow field exterior to the thin fiber is uniquely determined by imposing a fiber integrity condition: the surface velocity field on the fiber must be constant along cross sections orthogonal to the fiber centerline. Furthermore, a careful estimation of the residual, together with stability estimates provided by the PDE well‐posedness framework, allows us to establish error estimates between the slender body approximation and the exact solution to the above problem. The error is bounded by an expression proportional to the fiber radius (up to logarithmic corrections) under mild regularity assumptions on the 1D force density and fiber centerline geometry. © 2019 Wiley Periodicals, Inc.

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.

Chapter 15 - Compressible Flow

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.

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.

Automatic detection of oblique shocks and simple waves in schlieren images of two-dimensional supersonic steady flows

We construct analytical solutions for the problem of nonlinear supersonic flow past slender bodies of revolution due to small amplitude oscillations. The method employed is based on the splitting of the time dependent small perturbation equation to a nonlinear time independent partial differential equation (P.D.E.) concerning the steady flow, and a linear time dependent one, concerning the unsteady flow. Solutions in the form of three parameters family of surfaces for the first equation are constructed, while solutions including one arbitrary function for the second equation are extracted. As an application the evaluation of the small perturbation velocity resultants for a flow past a right circular cone is obtained making use of convenient boundary and initial conditions in accordance with the physical problem.

The reverse-flow theorem is extended to the case of a wing mounted on a cylindrical body of any cross-sectional shape. Also, two TrefTtz-plane theorems relating the total lift and rolling moment on such a body to the corresponding values for the wings are derived. Based on these results, some further general relationships between integrated lifts and moments on the wing, the body, and the combination are obtained. I t is shown that the lift distribution on the wing-body combination in reverse flow is the influence function for lift of the combination. Further, it is shown that the influence function for rolling moment is the lift distribution over the combination rolling at constant angular velocity in reverse flow. I t is also shown tha t the lift induced on the wings by a unit body angle of attack is equal to the lift induced on the body by a unit wing angle of attack in reverse flow. In lifting-line theory, slender-body theory, or for any wings having a span wise axis of symmetry (such as rectangular or diamond wings), the need for consideration of reverse flow is eliminated, and the results assume a particularly simple form. The results are valid for subsonic or supersonic flow within the limitations of linearized theory.

Erosive burning rate studies of composite propellants under high cross flow velocities

Propulsion system design with smart vortex generators

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.

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 a supersonic flow with a discontinuous surface of tangential velocity, a Mach wave from a supersonic source is reflected and refracted at the discontinuous surface. According to the image method, a reflected wave is expressed by an image source and a transmitted wave by a transmitted source. Further more, the image method is applied to reflection and transmission of a Mach wave in a supersonic flow with two different boundaries: a discontinuous surface and a solid wall. a lot of image and transmitted sources are introduced in a flow field as the interference between these boundaries. A flow disturbed by an aerofoil is discribed as a function of the original sources distributed on the upper and lower surfaces of an aerofoil and their image sources. We determined numerically the effect of two boundaries on the characteristics of a lens section and so on.

The sonic boom generated by a slender body of revolution aerodynamically shaded by another body is numerically investigated. The aerodynamic shadow is created by a disk placed upstream of the slender body across a supersonic free-stream flow. The disk size and its position upstream of the body are chosen in such a way that the aerodynamically shaded flow is quasi-stationary. A combined method of phantom bodies is used for sonic boom calculations. The method is tested by calculating the sonic boom generated by a blunted body and comparing the results with experimental investigations of the sonic boom generated by spheres of various diameters in ballistic ranges and wind tunnels. The test calculations show that the method of phantom bodies is applicable for calculating far-field parameters of shock waves generated by both slender and blunted bodies. A possibility of reducing the shock wave intensity in the far field by means of the formation of the aerodynamic shadow behind the disk placed upstream of the body is estimated. The calculations are performed for the incoming flow with the Mach number equal to 2. The effect of the disk size on the sonic boom level is calculated.