Preiously, we examined supersonic flow oer (sharp) concae corners/turns oblique shock allows flow to make this (compression) turn What happens if: turn is conex (expansion) already shown expansion shock impossible (entropy would be destroyed) turn is gradual (concae or conex) Prandtl eyer - Copyright 00 by Jerry. Seitzman. All rights resered. > > > > > >
Gradual turn is made up of large number of infinitessimal turns/corners Each turn has infinitessimal flow change each turn produced by infinitessimal waeach wae Flow is uniform and isentropic between each turn/corner length between each is arbitrary could be zero length (sharp turn) and waes collapse to one point Prandtl eyer - Copyright 00 by Jerry. Seitzman. All rights resered. > sin a a sin b sin a > sin sin
Problem gien upstream conditions () and turning angle () find downstream conditions () Goal Prandtl eyer Fan ach number relations (similar to shock relations) Equations usemass, momentum, energy conseration, ach number def n., state equations Assumptions steady flow, quasi-d, reersibleadiabatic (isentropic) Prandtl eyer -3 Copyright 00 by Jerry. Seitzman. All rights resered.
Approach begin with single ach wae that expands supersonic flow through an infinitessimal (differential) angle of magnitude d essentially using differential t n t n d control olume n ass/omentum Conseration using same type of approach as for oblique shocks (two momentum components: t, n) d d find lack of pressure gradient tangent to wae gies t constant across wae d Prandtl eyer -4 Copyright 00 by Jerry. Seitzman. All rights resered.
Use t constant t,upstream cos t, downstream t ( d) cos( d) ( d)( cos cosd sin sin d) d 0 d cos cos dsin dcos ddsin sin/ sin cos d d sin d cos / d / d n 0 d t d n d n d (VIII.) d Prandtl eyer -5 Copyright 00 by Jerry. Seitzman. All rights resered.
Prandtl eyer -6 School of Aerospace Engineering Copyright 00 by Jerry. Seitzman. All rights resered. Relate and d d tpg/cpg a RT a a da d d T dt d T T d d d ( ) d T dt d T dt 0 T dt const. T T o o o ( ) d d d ( ) [ ] d d d Energy Conseration
Prandtl eyer -7 School of Aerospace Engineering Copyright 00 by Jerry. Seitzman. All rights resered. Relate VIII. and last eqn. d d d (VIII.) d d d is change in ach number associated with d turn angle d d integrate d d Need finite angle, - and finite
Perform Integration d ( ) tan tan (VIII.3) So, gien ( - ) and could sole VIII.3 for Can not inert VIII.3 analytically ( f(, )) either use interatie (e.g., numerical or guessing) method or find as a function of and tabularize or graph solution Prandtl eyer -8 Copyright 00 by Jerry. Seitzman. All rights resered.
Prandtl eyer -9 Copyright 00 by Jerry. Seitzman. All rights resered. ( ) tan tan Want to find () [really ( )] for any need to choose (arbitrary) reference condition, i.e., pick an where 0 let s choose 0 at tan ( ) tan represents angle through which a sonic flow would hae to turn to reach analagous to table of h(t) really h(t)-h(t ref ) just chosen h(t ref )0 (VIII.4) is turn angle for to Appendix D (John) for.4, 5 -
To find gien and find (for gien ) from table get from - - look up in table to find To find T, p,... use isentropic flow relations since expansion is isentropic (no shock) e.g., T o const To T T T Prandtl eyer -0 Copyright 00 by Jerry. Seitzman. All rights resered.
Gien: Uniform ach flow of nitrogen at 300 K flows oer compound wall corner: two turns, 0 and 6 0 6 Find: and T after final turn Assume: N is TPG/CPG with.4, steady, adiabatic, no work, iniscid,. Prandtl eyer - Copyright 00 by Jerry. Seitzman. All rights resered.
Analysis: (class exercise) To find gien and. find (for gien ) from table. get from - 3. look up in table to find Prandtl eyer - Copyright 00 by Jerry. Seitzman. All rights resered. 0.00 6.38.60 4.4.75 44.69.90 47.79 3.05 50.7.0 6.66.6 4.64.76 44.9.9 47.99 3.06 50.90.0 6.93.6 4.86.77 45..9 48.9 3.07 5.09.03 7.0.63 4.09.78 45.33.93 48.39 3.08 5.8.04 7.48.64 4.3.79 45.54.94 48.59 3.09 5.46.50 39..65 4.53.80 45.75.95 48.78 3.0 5.65.5 39.36.66 4.75.8 45.95.96 48.98 3. 5.83.5 39.59.67 4.97.8 46.6.97 49.8 3. 5.0.53 39.8.68 43.9.83 46.37.98 49.37 3.3 5.0.54 40.05.69 43.40.84 46.57.99 49.56 3.4 5.39.55 40.8.70 43.6.85 46.78 3.00 49.76 3.5 5.57.56 40.5.7 43.84.86 46.98 3.0 49.95 3.6 5.75.57 40.74.7 44.05.87 47.9 3.0 50.4 3.7 5.93.58 40.96.73 44.7.88 47.39 3.03 50.33 3.8 53..59 4.9.74 44.48.89 47.59 3.04 50.5 3.9 53.9 6 3
Fan angle angle between first and last ach wae useful to determine when expansion has ended in flowfield for a gien distance away from wall From geometry Fan Angle Prandtl eyer -4 Copyright 00 by Jerry. Seitzman. All rights resered. ( ) ( ) ( ) ( ) (VIII.5) sin - Fan Angle sin
50 Examine plot of as function of - 00 50 0 max 30.45 0 0 0 30 Prandtl eyer -5 Example max tan Copyright 00 by Jerry. Seitzman. All rights resered. ( ) tan.4 p 0 max 30.45 6.38 6 max 30.45 84.96 04. 45.5 As increases, reach maximum turn angle ( max ~30.5 for.4) So as increases, max. angle flow can turn ( max ) decreases P <04.
Analytic Expression For : max max Prandtl eyer -6 tan 90 30.45 59. 08.5 Copyright 00 by Jerry. Seitzman. All rights resered. 90 5/3.4.3. ( ) tan ;tan (VIII.6) ( ) 90 As decreases (higher temperatures, bigger molecules), maximum turn angle increases max smaller in real flows T and p drop through turn condensation of gas nonequilibrium flow
Already showed that it does not matter if expansion turn is sharp or smooth > still get same solution, P- faninfinite set of ach waes unless we exceed the maximum turning angle, final properties just function of total turn angle smooth turn just means expansion process takes place oer longer distance > Prandtl eyer -7 Copyright 00 by Jerry. Seitzman. All rights resered.
What happens if we hae a smooth concae turn? Prandtl eyer -8 since flow direction change is small, can still get set of weak ach waes Prandtl-eyer compression: Copyright 00 by Jerry. Seitzman. All rights resered. > < ( ), so <0 howeer unlike expansions, compressions merge together they coalesce to form oblique shock flow that went through P compression is isentropic, outer flow has entropy rise (p o loss) size of P region depends on and curature