wilson t, wan wu, and jesus h project me404

A Uniform Supersonic Stream Project

ME404

Project 1

CalState La

Wilson Tong, Wan Wu, and Jesus GonzalesJune 4,2015

CONTENTS

List of Figures 1

References 1

1 INTRODUCTION 1

2 PROBLEM DESCRIPTION 2

3 MATHEMATICS 23.1 The Mach angle . . . . . . . . . . . . . . . . . . 2

3.1.1 Steady Flow . . . . . . . . . . . . . . . 23.1.2 Mn1 . . . . . . . . . . . . . . . . . . . . 23.1.3 Calorically Perfect Gas . . . . . . . . . . 23.1.4 Mach 2 . . . . . . . . . . . . . . . . . . 33.1.5 Θ−β −MRelation . . . . . . . . . . . . 3

4 RESULT AND DISCUSSION 34.0.1 Calculated data table . . . . . . . . . . . 3

5 CONCLUSION 5

List of Figures1 Concave corner . . . . . . . . . . . . . . . . . . 22 Convex corner also known as Expansion Fan . . . 23 Cacluated Results of Problem . . . . . . . . . . . 34 Shockwave and Deflection . . . . . . . . . . . . 45 P2 and P1 . . . . . . . . . . . . . . . . . . . . . 46 Po2 and P2 . . . . . . . . . . . . . . . . . . . . 47 To2 and Mach . . . . . . . . . . . . . . . . . . . 48 Mach2 and Delta . . . . . . . . . . . . . . . . . 5

Proceedings of Compressible AerodynamicsME 404

June 4, 2015, Los Angeles, California, USA

Wilson Tong∗

Editor, Group Member of ProjectME Undergraduate

Email: [email protected]

Wan Wu†



Jesus Gonzalez‡



ABSTRACTIn this Project we have conducted a study via to numeri-

cal simulations of a uniform supersonic stream with a the givenMach number of with a certain pressure and the temperature inkelvins. [?] This encounters a compression corner which deflectsthe stream by an certain angle. A uniform supersonic flow isbounded on one side by a surface through a point of the surfaceis then deflected upward through an angle and flow streamlinesare deflected upward, toward the main bulk of the flow abovethe surface. The mathematical model of the uniform supersonicstream is consist upon the continuity equation along with the re-lations of oblique shocks. With the given Mach, temperature,and pressure; calculating the shock wave angle and p2, T2, M2,To2, and Po2 behind the shock wave can be repeated for differentMach, pressure, and angle, but without the given Mach, pressureand angle we cannot solve the different pressure, temperature,Mach two, Temperature knot two, Pressure knot two.

NOMENCLATUREc Speed of sound in air [m/s]k The ratio of specific heatM1 Mach Number of 1M2 Mach Number of 2P1 Pressure of the Fluid referring to Mach 1 [atm]Po1 Pressure knot of the Fluid referring to Mach 1 [atm]

∗Address all correspondence related to ASME style format and figures to thisauthor.

†Address all correspondence related to ASME style format and figures to thisauthor.

‡Address all correspondence related to ASME style format and figures to thisauthor.

P2 Pressure of the Fluid referring to Mach 2 [atm]Po2 Pressure knot of the fluid referring to Mach 2 [atm]T 1 Temperature of the fluid referring to Mach 1 [K]To1 Temperature knot of the fluid referring to Mach 1 [K]T 2 Temperature of the fluid referring to Mach 2 [K]To2 Temperature knot of the fluid referring to Mach 2 [K]v Velocity of the fluid [m/s]Greek symbolsΘ The angle of the streamρ Density of fluid [g/m3]β shock wave angle (degree angle)µ Mach Angle

1 INTRODUCTIONUniform supersonic stream is a Oblique shock with expan-

sion waves that can usually occur when supersonic flow is turnedinto itself as you can see in fig 1. [?]The uniform supersonicstream is bounded on one side by a surface and the surface isdeflecting upward through an angle. The flow streamlines aredeflected upward, toward the main bulk of the flow above the sur-face. This flow in direction take place across a shock wave whichis oblique to any free-stream direction. All flow stream-linesexperience the same deflection angle at the shock, but the flowdownstream of the shock is uniform and parallel which followsthe direction of the wall down-stream across the shock wave. Asthe Mach number decreases the pressure, temperature, and den-sity increases.

When supersonic flow is turned away from itself an expan-sion wave is then formed. [?] The surface is deflected downwardthrough an angle and deflected downward away from the mainstream of flow above the surface. Which causes the surface to

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change in flow of direction that takes place across an expansionwave. The flow streamlines are smoothly curved through theexpansion fan until they are all parallel to the wall behind thepoint as seen in Fig 2. All properties through the expansion wavechange smoothly and continuously with the exception of the wallstreamline which changes discontinuously across the expansionwave as the Mach number increases and the pressure, tempera-ture, and density decreases.

FIGURE 1: Concave corner

FIGURE 2: Convex corner also known as Expansion Fan[?]

2 PROBLEM DESCRIPTIONA uniform supersonic stream with

M1= 3, p1= 1 atm, and T1= 288K encounters a compressioncorner which deflects the stream by an angle θ = 20◦

Calculate the shock wave angle P2, T2, M2, To2 and Po2behind the shock-wave. Repeat the problem forM1 = 0, 1, 1.5, 2, 3, 10P1 = 1, 2, 3, 4, 5 [atm]θ = 0,10,20,45,60,90

3 MATHEMATICS3.1 The Mach angle

µ = arcsin1M

(1)

3.1.1 Steady Flow with no body forces the tangentialcomponent becomes

−rho1u1)ω1+(rho2u2)ω2 = 0 (2)

We find that both cancel out and becomes

ω1 = ω2 (3)

3.1.2 Mn1 Normal component of the upstream MachNumber Mn1 for oblique shock wave

Mn1 = M1sin(β ) (4)

3.1.3 Calorically Perfect Gas

ρ2ρ1

=(k+1)Mn12

(k−1)Mn12 +2(5)

P2P1

= 1+(2k

k+1)(Mn12 −1) (6)

Mn22 =Mn12 +2( 2

k−12k

k−1 (Mn12 −1)(7)

T 2T 1

=P2P1

ρ2ρ1

(8)


3.1.4 Mach 2 When M is greater-than 1, M2 can befound from Mn2 and the geometry in figure 1. [?]

M2 =Mn2

sin(β −θ)(9)

3.1.5 Θ−β −MRelation

tanθ = 2cotβ (M12sin2β −1

M12(k+ cos2β )+2) (10)

With equation 10, we can use to find the relation betweentheta-beta-M. beta can be found through number of iteration,where equation 10 becomes zero. For every given M1 there isa maximum deflection angle theta max. [?] If the angle theta isgreater than theta max then equation 10 becomes invalid. [?] Iftheta is 0 then beta = pi/2 this would correspond to normal shockwave. [?] For fixed deflection angle theta, as the free-streamMach number decreases from high to low supersonic value, thewave angle increase. [?]

4 RESULT AND DISCUSSION

FIGURE 3: Cacluated Results of Problem

4.0.1 Calculated data table This table consist of allthe calculated results. Through calculation we found that whendelta is 0, the shock wave angle is undefined and also for delta90 it can never occur because an oblique shock must be an acuteangle. We have calculated must of the data, some data remainN/A because this value in invalid for a oblique shock.


FIGURE 4: Shockwave and Deflection

In Figure 4, is an incomplete due to the amount of dataused to analysis. In general for a oblique shock wave. As deltaincreases the Mach 2 decreases while the shock wave anglealso increases. If delta increases beyond delta max the shockwave become detached. [?] Reversely when delta decreasethe Mach 2 increases with shock wave angle decreasing. Thisbehavior is known as weak shock case. Therefore, in Figure 4when deflection reach 45 and greater the shock wave becomesdetached.

FIGURE 5: P2 and P1

In an oblique shock, P2 is always greater than P1. FromFigure 5 graph shows that P2 linearly increasing respect to P2when Mach 1 number is increasing.

This behavior is very similar to Figure 6: Where Po2 influ-ence by Mach 2 number.

FIGURE 6: Po2 and P2

For the graph shown below:

FIGURE 7: To2 and Mach

This result shows the behavior of the oblique shock in theweak case, as Mach 1 increases T2 also increases. Furthermore,


Figure 7, shows that temperature change corresponding to Machnumber.

The graph below shows the relation of M2 and delta withrespect to the increasing Mach number.

FIGURE 8: Mach2 and Delta

In Figure 8, shows M2 with respect to delta base on the in-creasing of Mach number. When the Mach number is at 1 or1.5 with respect to delta 20, it shows that the M2 very high dueto transient flow. As result for the Mach number 2 and 10, asdelta increases the M2 decrease which how oblique wave shouldbehave in a weak shock case.

5 CONCLUSIONTo conclude, we general have a good understanding how

oblique shock wave behave in respect to the change of delta,Mach1, P1, T2 and wave angle. To have better results for the fu-ture we require to have more data value to obtain a better graph.To have more accurate values for to computer M2, we must havea smaller iteration process.

REFERENCES[1] A. John D, Modern Compressible FLow with Historic Per-

spective. New York: Mc Graw Hill, 2012.[2] D. V. Kulkami, “Compressible flow

two dimensional analysis.” [Online]. Available:http://nptel.ac.in/courses/101103004/22


wilson t, wan wu, and jesus h project me404

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