Schlieren Images: An Analytical Comparison of Shock Wave Angles Obtained from Various Methods

Abstract

This project’s aim is to demonstrate the validity of the shock wave angles gained from several methods to the shock wave angles gained from Schlieren images. The Solutions covered will be a Computational Fluid Dynamics Solution, a Solution method using a graph of the theta – beta – Mach relation, and an Analytical Solution with an average percentage difference of 10.1505%, 1.59%, and 1.6054% respectively.

Nomenclature

Introduction

Background

Schlieren images

Schlieren images are images of objects that display the density variations in the transparent media around that object. Schlieren images are images that display density variations in transparent media. The light intensity distribution portrays the expansions and compressions of flow. The brighter regions correspond to low-density flow while the darker regions correspond to high-density flow. These images are incredibly useful for capturing information about compressible flow at high speeds due to the change in density of a compressible medium through a shock wave.

Technology background

This form of technology has existed long before August Toepler came up with the term in 1859. Throughout time, humanity is steadily increasing its technological capability. Now, the vast amounts of computing power that is accessible to any given individual are increasing exponentially. The Navier stokes equations can now be easily simulated with readily available open-source software. However, it is important to verify a computational fluid dynamics (CFD) simulation with real-world experimental results to determine the validity of the simulated results because the exact properties of the flow at any given point are not certain. CFD only gives a close depiction of the truth.

Theory

Figure 1: The change in compressible flow with change in velocity.

Figure 2: Example of the geometry of an oblique shock wave in a 2D plane

Upon evaluation of both regions 1 and 2 in Figure 2 by using the conservation of the momentum equation and separating the velocity into u and w components where u is normal and w is tangent to the shock wave, it can be found that

Experimental Data

The angles can be found in the Schlieren images using various online protractors. I used root cad pro to measure the corner and shock wave angles for each image.

Measuring the angles from the Schlieren Images

Figure 3: Methods on measuring the angles from the CFD simulation

A green horizontal represents the start of the corner half-angle, theta. Measuring the angle from the green line to the top red line will result in the corner half angle. Measuring from the green line to the top dark blue line will result in the shock wave angle, beta.

CFD Simulation

Geometry setup

The shape of geometry does not necessarily matter for this scenario as the objective is to simulate Mach flow over a wedge with varying corner angles of 5, 10, and 15 degrees.

Figure 4: Geometry setup for a theta angle of 15 degrees, H46 = 0. 5mm, V49 = 0.98 mm, H43 = 1.5 mm, and V42 = 1.25 mm.

Mesh Setup

Face meshing was used to evenly distribute the quadrilateral shape evenly along with the geometry. The nominal element size used for each mesh was 0.0008 m in length and width. The wedge’s boundary was labeled as wedge. The boundary before that was labeled as symmetry. The Pressure far-field boundary condition was set to the rest of the boundaries.

Figure 5 depicting a mesh of the previous geometry with elements of 0.0008 m in size.

Fluent Setup

The temperatures for each case were all set to 300 Kelvin. The operating pressures for each case were all set to 0 Pascals. The Viscous model used is inviscid and density-based. The medium used to simulate flow is air treated to be an ideal gas The Pressure Far Field boundary condition allows modeling of the free stream compressible flow coming from infinity and uses Riemann invariants to find flow characteristics at the flow boundary. The convergence criteria for each case are set to 1e-06.

Using the Graph to Find the Shock Wave Angles

One can find the shock wave angle beta by using the graph of the theta beta m relation provided.

Figure 6: Display of the curve

Calculation of the Shock Wave Angles using the Analytical Solution given in Rudd and Lewis

In an AIAA journal from 1998, Rudd and Lewis discuss an analytical solution for determining the shock wave angle from the theoretical equations above. An undergraduate from Maryland University, Chris Plumley created a MATLAB function that covers this solution in 2011. Their Solution is presented below:

Results

Table 1: Comparison of the results obtained through the various methods mentioned

Figure 7: Display of the curve created from the results

Table Comparison

Graphed V.S. Calculated

Graphed V.S. Simulated

Simulated V.S. Calculated

Discussion of Results Obtained from the Shlirein Images

There is some uncertainty in measuring the shock wave angles directly from the images. However, these angles are the truest out of all of the other methods because their basis is in reality compared to angles whose basis is ideal.

Discussion of Results Obtained from the CFD

The results coming from CFD for determining the shock wave angle were very clear and gave a massive amount of information about the state equations that aren't necessary to the scope of this project. Each case converged to the convergence criteria within a couple of minutes. I decided to pick the line that was in the middle of the shock wave for determining the shock wave angle. Utilizing Root CAD Pro along with the images rendered from the CFD simulation gave a very precise measurement of the shock wave angles for each case. Mesh refinement was not necessary because it would blur the transition lines of where the shock wave angles would be.

Discussion of Results Obtained from the Graph

The shock wave angles determined from the graph will have some uncertainty due to grid increment sizing not being small enough to match the exact beta angle. This is expected because there will always be uncertainty with precisely determining values from this method.

Discussion of Results Obtained from Calculated Values

The shock wave angles that were analytically determined, if rounded, would exactly match the results coming from the CFD simulation and somewhat match the results coming from the graph. This serves to add validity to each of these methods for these cases. However, this event might not always occur with higher Mach numbers.

References

MATLAB. (2010). version 7.10.0 (R2010a). Natick, Massachusetts: The MathWorks Inc.

ANSYS (2016). ANSYS Fluent - CFD Software | ANSYS

W. Ethan Eagle (2021). Theta Beta Mach Analytic Relation (https://www.mathworks.com/matlabcentral/fileexchange/32777-theta-beta-mach-analytic-relation), MATLAB Central File Exchange. Retrieved November 3, 2021.

Sambit Supriya Dash (2021). Theta-Beta-MachNo Relation (Plot) for Oblique Shock Waves (https://www.mathworks.com/matlabcentral/fileexchange/72590-theta-beta-machno-relation-plot-for-oblique-shock-waves), MATLAB Central File Exchange. Retrieved November 3, 2021.

Anderson. (2016). Fundamentals of aerodynamics (6th ed.). McGraw-Hill Education.

Narsipur (2021). Project 1 – Shock Wave Analysis, Retrieved from https://msstate.instructure.com/courses/55907/files/4824831?module_item_id=1267915

RootPro Co., Ltd. (2018). RootPro CAD. Version 10.10. Akaho, Komagane-shi, Nagano-ken, 399-4117, Japan. Microsoft Store

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