Analytical Shock Standoff and Shape Prediction with Validation for Blunt Face Cylinder

Abstract

The paper presents the details and validation for an analytical model for predicting shock stand-off distance from a blunt body at supersonic speeds. Comparisons are made to experimental data for Mach Numbers 1.2 to 6.0 with flat face objects. Additionally a method to predict the oblique shock shape is presented and qualitative comparisons are made to both experimental and computational data. The usefulness of such a predictive tool set provides help in interpreting experimental data such as Schlieren images and in capturing a shock location using computational methods with the initial grid comforming to the shock shape when known apriori. An example of such usage is given to help interpret a poor quality Schlieren from a Mach 3.0 test. Nomenclatured = Cylinder diameter, angular distance between two circular sides of frustum e = Distance from control volume frustum to center of oblique shock wave h = Distance of height of control volume frustum starting from flat face object M1 = Mach Number upstream of oblique shock wave M2 = Mach Number immediately downstream of oblique shock wave at center Me = "exit" Mach Number aft of oblique shock wave expanding around blunt face Mi = Average Mach Number in ring segment of frustum P1 = Pressure upstream of oblique shock wave P2 = Pressure immediately downstream of oblique shock wave at center Pe = Pressure at exit Mach Number equal to unity Pi = Average pressure in ring segment of frustum R1 = Radius of control volume frustum R2 = Radius of blunt cylindrical geometry and second radius of control volume frustum T1 = Temperature upstream of oblique shock wave T2 = Temperature immediately downstream of oblique shock wave at center Te = Temperature at exit Mach Number equal to unity Ti = Average temperature in ring segment of frustum h = Height φ = Angle of curvature from shock tangent line to the intersection of the sonic line θ = Angle sonic line makes with surface geometry ρ1 = Density upstream of oblique shock wave 1 Retired, AIAA Retired Member. AIAA SciTech ρ2 = Density immediately downstream of oblique shock wave at center ρi = Average density in ring segment of frustum ψ = Shock wave cone angle δ = shock standoff distance

Numerical simulation of hypersonic flow with non-equilibrium chemical reactions around sphere

“…However, this approach is rather approximate, and finding the average density in the shock layer requires rather laborious calculations [46]. The model proposed in [45] takes into account the geometry of the aircraft head and flow quantities.…”

Shock-conforming mesh generation for aerodynamic analyses at supersonic regimes

“…Alternatively, the shock 25 structure could be estimated through analytical approaches, but the existing ones are limited to very specific geometries and flow conditions [5]. Examples of 2 this are the work of Catalano[6], defining the far field boundary of a CFD simulation of the Vega launcher using Billig's [7] approximate solution for bow shock shape and stand off distance for spheres and cylinders in supersonic flow; or 30 Carter [8], applying mesh alignment and refinement along the free stream Mach angle for sonic boom signature predictions.…”

“…The extensive adoption of CFD throughout the entire design process is nevertheless still limited by the high computational resources 5 that are often required to solve complex 3D flows over elaborated vehicle's geometric configurations. A lot of efforts are therefore continuously put forward by researchers and engineers to streamline and certify as much as possible the CFD process in order to reduce uncertainty and improve the design.…”

A Semi-empirical Model to Predict the Attached Axisymmetric Shock Shape

“…For calculating shock shape over wide ranges of Mach number and nose shape, the method of characteristics, though being laborious, is still a popular tool. However, in recent years, with the advent of modern processors, the computational methods as well as the experimental surveys are widely used to determine the shock shape and its stand-off distance for the bow and detached shocks [7][8][9][10].…”