sin The deflection angle … 2 A similar design was used on the F-14 Tomcat (the F-14D was first delivered in 1994) and achieved a maximum speed of Mach 2.34. β NOTE ON MAXIMUM SHOCK DEFLECTION* By GARRETT BIRKHOFF and JOHN W. WALSH (.Harvard University) The angle X = \{M) of maximum shock deflection, for a given Mach number M of flow, is of interest in various applications. cos + 2 Strong solutions may be observed in confined geometries (such as inside a nozzle intake). Tabulated Values: Inputs. Two-dimensional supersonic flow past a wedge can be described bearing the shock polar in mind. 2 γ See also bow shock or oblique shock; Such a shock occurs when the maximum deflection angle is exceeded. 1 ρ The gas is assumed to be ideal air. Concorde (which first flew in 1969) used variable geometry wedge-shaped intakes to achieve a maximum speed of Mach 2.2. The calculator computes ratios to free stream values across an oblique shock wave, turn angle, wave angle and associated Mach numbers (normal components, M n, of the upstream). Early supersonic aircraft jet engine intakes were designed using compression from a single normal shock, but this approach caps the maximum achievable Mach number to roughly 1.6. This information can in turn be used for design and control purposes. It gives the critical angle for attached shocks past a wedge [2, p. 53], and that for jetless wedge collapse [3]. − sin ≈ 2 1 Within the θ-β-M equation, a maximum corner angle, θ MAX, exists for any upstream Mach number. γ 2 . − M ) 1 1 When θ > θ MAX, the oblique shock wave is no longer attached to the corner and is replaced by a detached bow shock. Examples: Space return vehicles (Apollo, Space shuttle), bullets, the boundary of a magnetosphere. γ Using the continuity equation and the fact that the tangential velocity component does not change across the shock, trigonometric relations eventually lead to the θ-β-M equation which shows θ as a function of M1 β, and É£, where É£ is the Heat capacity ratio. + = The shock wave then sets itself up at the angle given by the weaker shock. The required input is the Mach number of the upstream flow and the wedge angle. 1 {\displaystyle {\frac {\rho _{2}}{\rho _{1}}}={\frac {(\gamma +1)M_{1}^{2}\sin ^{2}\beta }{(\gamma -1)M_{1}^{2}\sin ^{2}\beta +2}}}, T 2. 1 ⁡ M 1 M Oblique shocks also occur downstream of a nozzle if the expanded pressure is different from free stream conditions. For each angle of attack, calculate the oblique shock angles on the upper (βu) and lower (βl) wedge surfaces. β = 2 + ⁡ The shock polar constitutes a graphical tool that can be used to analyze and understand a oblique shock situation. M Unlike after a normal shock where M2 must always be less than 1, in oblique shock M2 can be supersonic (weak shock wave) or subsonic (strong shock wave). The most common way to produce an oblique shock wave is to place a wedge into supersonic, compressible flow. Weak solutions are often observed in flow geometries open to atmosphere (such as on the outside of a flight vehicle). {\displaystyle {\frac {T_{2}}{T_{1}}}\approx {\frac {2\gamma (\gamma -1)}{(\gamma +1)^{2}}}M_{1}^{2}\sin ^{2}\beta . p ⁡ γ Maximum deflection angle. Placing a diamond-shaped object at an angle of attack relative to the supersonic flow streamlines will result in two oblique shocks propagating from the front tip over the top and bottom of the wing, with Prandtl-Meyer expansion fans created at the two corners of the diamond closest to the front tip. 1 p The issue of maximum deflection has a practical application aside from the obvious configuration used as a typical simple example. The deflection angle, delta, for a given wave angle, theta, and a given Mach number M is governed by. When θ > θ MAX, the oblique shock wave is no longer attached to the corner and is replaced by a detached bow shock. Results from running this program. ( Source code for the maximum deflection angle program. 1 1. 2 T ( For a perfect atmospheric gas approximation using γ = 1.4, the hypersonic limit for the density ratio is 6. the curve length L is equal to 100 * delta angle/2 but if you only know the deflection angle then use R = 5729. Only two properties need to be specified in order to define uniquely a given oblique shock wave. The total flow after the oblique shock can also be supersonic, which depends on the boundary layer and the deflection angle. {\displaystyle M_{2}={\frac {1}{\sin(\beta -\theta )}}{\sqrt {\frac {1+{\frac {\gamma -1}{2}}M_{1}^{2}\sin ^{2}\beta }{\gamma M_{1}^{2}\sin ^{2}\beta -{\frac {\gamma -1}{2}}}}}.}. YËûMÀ¿_єÁ®joÀè݅҉'g7„N¦B;|¼$“ ZŽÃã ±È2K¢ßö ¬ -@ÂJŬ&5{_PÎú}\4kÄ«”OíNÛHD°/ÙuóÚ²}•]1ðìÊ8r8Vób`©…û. When θ > θ MAX, the oblique shock wave is no longer attached to the corner and is replaced by a detached bow shock. ⁡ Within the θ-β-M equation, a maximum corner angle, θ MAX, exists for any upstream Mach number. ( There is a maximum turning angle θmax for any given upstream Mach number M1. M 2 β γ The influence of a uniform magnetic field on the maximum flow deflection through oblique magnetohydrodynamic shock waves is investigated under the assumptions, that (1) the fluid medium is an infinitely conducting gas, and (2) the applied magnetic field is aligned with the incident stream. 2 {\displaystyle {\frac {\rho _{2}}{\rho _{1}}}\approx {\frac {\gamma +1}{\gamma -1}}.}. This can be attributed to the fact that using one or a combination of oblique shock waves results in more favourable post-shock conditions (smaller increase in entropy, less stagnation pressure loss, etc) when compared to utilizing a single normal shock. {\displaystyle {\frac {p_{2}}{p_{1}}}\approx {\frac {2\gamma }{\gamma +1}}M_{1}^{2}\sin ^{2}\beta }, ρ It is found that for fixed values of the upstream Laval number the maximum deflection angle … ρ + }, M 2 A θ-β-M diagram, common in most compressible flow textbooks, shows a series of curves that will indicate θMAX for each Mach number. 2 The shock­wave angle of 30° is not much larger than д; the deflection angle of 6.5° is also small—consistent with the relative weakness of the shock wave. Pressure, Mach number and Wave Angle changes through oblique shock waves. (a) Determine the maximum angle of attack, αcrit, that is possible for attached oblique shocks to exist on the upper and lower surfaces. 1 Within the θ-β-M equation, a maximum corner angle, θ MAX, exists for any upstream Mach number. For a given Mach number, M1, and corner angle, θ, the oblique shock angle, β, and the downstream Mach number, M2, can be calculated. sin β On the slide, a supersonic flow at Mach number M approaches a shock wave which is inclined at angle s. The flow is deflected through the shock by an amount specified as the deflection angle - a. 1 − The shock wave angles (beta) are plotted for range of flow deflection angles (theta) varying with specific upstream supersonic Mach numbers (M1). Oblique Shock Relation¶ An oblique shock results from a sudden change of direction of supersonic flow. sin ( Within the θ-β-M equation, a maximum corner angle, θ MAX, exists for any upstream Mach number. Strong solutions are required when the flow needs to match the downstream high pressure condition. Oblique Shock Calculations This form calculates properties of air flow through an oblique shock wave. 1 Similar to a normal shock wave, the oblique shock wave consists of a very thin region across which nearly discontinuous changes in the thermodynamic properties of a gas occur. ≈ cot δ = tan θ [(γ + 1) M 2 2 (M 2 sin 2 θ-1)-1] This is Eq. − In other words, the deflection angle, , defined in Figure 15.1, is positive. The normal shock analysis dictates that after the shock, the flow is always subsonic. (4 Points) Show That The Maximum Deflection Angle For An Attached Oblique Shock Wave Can Be Written As: V Nl Omax = Arctan - Arctan V Vn2 Vni Where Vni And Vn2 Are The Normal Velocities Upstream And Downstream Of The Shock, Respectively. M The deflection angle, $\delta$, is the direction of the flow after the shock (parallel to the wall). 1 Maximum deflection angle. Many supersonic aircraft wings are designed around a thin diamond shape. Weak shocks are more common to be found in nature. 2 Strong solutions are required when the flow needs to match the downstream high pressure condition. sin γ = While the upstream and downstream flow directions are unchanged across a normal shock, they are different for flow across an oblique shock wave. The maximum value of theta for particular mach numbers are plotted for the corresponding beta value. 2 + 2 8 --Centerbody Deflection Angle - Oblique Shock Angle In this note, we formulate a closed-form solution for the oblique shock angle, ct, in terms of the free-stream Mach number, M1, the eenterbody deflection angle, 8, and the ratio of the specific heats, 7. . Solution Assumptions: Air is an ideal gas with a constant specific heat ratio of k = 1.4. The upstream streamlines are uniformly deflected after the shock wave. M 1 + mu above is the Mach angle = arcsin(1/Mach). 1 β The transition between them is indicated by a long‐dashed curve and corresponds to the detachment angle θ d. It is always possible to convert an oblique shock into a normal shock by a Galilean transformation. γ 2 sin Once, the inclination angle reaches the "maximum potential energy,'' a change in the flow direction is no longer possible. For Ma = 5, we read from curve, Minimum shock (or wave) angle: = 12° Maximum deflection (or turning) angle: = 41.5° Next: Oblique Shock Examples Up: Detached Shock Previous: Detached Shock Index Issues Related to the Maximum Deflection Angle. (4 points) Show that the maximum deflection angle for an attached oblique shock wave can be written as: where vn1 and vn2 are the normal velocities upstream and downstream of the shock, respectively. β ) ρ − − 138 of NACA report 1135 and is illustrated in the following chart. sin {\displaystyle {\frac {p_{2}}{p_{1}}}=1+{\frac {2\gamma }{\gamma +1}}(M_{1}^{2}\sin ^{2}\beta -1)}, ρ The first has larger shock angle than the second. Within the θ-β-M equation, a maximum corner angle, θMAX, exists for any upstream Mach number. 1 Maximum deflection angle. If the wall angle exceeds this, or θ > θmax, no oblique shock is possible. In order to check that the function is coded correctly, I have made a little program that calls MaxRampAngle at the Mach numbers in the table on the previous page. sin γ 2 1 Figure 15.1: An oblique shock. γ 2 In the typical example, a wedge or a cone moves into a still medium or gas flows into it. The solid curves denote weak shock solutions, and the short‐dashed curves denote the strong shock solutions. The rise in pressure, density, and temperature after an oblique shock can be calculated as follows: p ⁡ γ ⁡ β γ (b) Select five angles of attack ranging between α = 0 and α = αcrit. The first range is when the deflection angle reaches above the maximum point. An oblique shock wave is a shock wave that, unlike a normal shock, is inclined with respect to the incident upstream flow direction. Oblique shocks are often preferable in engineering applications when compared to normal shocks. When θ > θ MAX, the oblique shock wave is no longer attached to the corner and is replaced by a detached bow shock. sin 1 1 . {\displaystyle {\frac {T_{2}}{T_{1}}}={\frac {p_{2}}{p_{1}}}{\frac {\rho _{1}}{\rho _{2}}}. 2 1 ) Free online beam calculator for generating the reactions, calculating the deflection of a steel or wood beam, drawing the shear and moment diagrams for the beam. - For each Mach number the deflection angle has a maximum value past which the oblique shock detaches from the body and forms a bow shock. 2 . 2 A detached shock is commonly seen on blunt bodies, but may also be seen on sharp bodies at low Mach numbers. The pressure and density ratios can then be expressed as: p [1], tan ) 2 ≈ 2 2 γ Strong solutions may be observed in confined geom… p ) {\displaystyle \tan \theta =2\cot \beta {\frac {M_{1}^{2}\sin ^{2}\beta -1}{M_{1}^{2}(\gamma +\cos 2\beta )+2}}}. 1 ( The weak shock is almost always seen experimentally. ( Maximum deflection angle. 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