A uniform flow at atmospheric pressure having M1 = 3.0 is deflected by 20°. What are the Mach number and pressure in the flow after the deflection if it occurs through (a) an oblique shock wave from a compression corner (Figure 15.23 right side panel), (b) an isentropic compression from a curved

Question

A uniform flow at atmospheric pressure having [latex]M_1=3.0[/latex] is deflected by 20°. What are the Mach number and pressure in the flow after the deflection if it occurs through (a) an oblique shock wave from a compression corner (Figure 15.23 right side panel), (b) an isentropic compression from a curved wall (Figure 15.25a), and (c) an isentropic expansion (Figure 15.25b).

Accepted Answer

For (a) an oblique shock wave must be considered. Using Figure 15.22, [latex]M_1=3[/latex] and δ = 20°, leads to σ = 37.5°, so [latex]M_1\sin\sigma=1.83[/latex]. Thus, from (15.61) and (15.63):

[latex]\frac{p_2}{p_1}=1+\frac{2 \gamma}{\gamma+1}\left[M_1^2 \sin ^2 \sigma-1 ight][/latex], (15.61)

[latex]M_2^2 \sin ^2(\sigma-\delta)=\frac{(\gamma-1) M_1^2 \sin ^2 \sigma+2}{2 \gamma M_1^2 \sin ^2 \sigma+1-\gamma}[/latex]. (15.63)

[latex]p_2=\frac{p_2}{p_1} p_1=\left(1+\frac{2 \gamma}{\gamma+1}\left[M_1^2 \sin ^2 \sigma-1 ight] ight)(1.0 atm )=3.74 atm[/latex]., and

[latex]M_2=\frac{1}{\sin (\sigma-\delta)}\left[\frac{(\gamma-1) M_1^2 \sin ^2 \sigma+2}{2 \gamma M_1^2 \sin ^2 \sigma+1-\gamma} ight]^{1 / 2}=2.03[/latex].

For (b), the Prandtl-Meyer function may be used. Here the initial flow angle is 0° and [latex] u\left(M_1=3 ight)=[/latex] 49.76°. Thus, [latex] u\left(M_2 ight)=0-20^{\circ}+49.76=29.76^{\circ}[/latex], for which [latex]M_2[/latex] = 2.125. The downstream pressure can be recovered from the Table 15.1:

TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)
M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*
0	1	1	1	∞	0.62	0.7716	0.831	0.9286	1.1656
0.02	0.9997	0.9998	0.9999	28.9421	0.64	0.7591	0.8213	0.9243	1.1451
0.04	0.9989	0.9992	0.9997	14.4815	0.66	0.7465	0.8115	0.9199	1.1265
0.06	0.9975	0.9982	0.9993	9.6659	0.68	0.7338	0.8016	0.9153	1.1097
0.08	0.9955	0.9968	0.9987	7.2616	0.7	0.7209	0.7916	0.9107	1.0944
0.1	0.993	0.995	0.998	5.8218	0.72	0.708	0.7814	0.9061	1.0806
0.12	0.99	0.9928	0.9971	4.8643	0.74	0.6951	0.7712	0.9013	1.0681
0.14	0.9864	0.9903	0.9961	4.1824	0.76	0.6821	0.7609	0.8964	1.057
0.16	0.9823	0.9873	0.9949	3.6727	0.78	0.669	0.7505	0.8915	1.0471
0.18	0.9776	0.984	0.9936	3.2779	0.8	0.656	0.74	0.8865	1.0382
0.2	0.9725	0.9803	0.9921	2.9635	0.82	0.643	0.7295	0.8815	1.0305
0.22	0.9668	0.9762	0.9904	2.7076	0.84	0.63	0.7189	0.8763	1.0237
0.24	0.9607	0.9718	0.9886	2.4956	0.86	0.617	0.7083	0.8711	1.0179
0.26	0.9541	0.967	0.9867	2.3173	0.88	0.6041	0.6977	0.8659	1.0129
0.28	0.947	0.9619	0.9846	2.1656	0.9	0.5913	0.687	0.8606	1.0089
0.3	0.9395	0.9564	0.9823	2.0351	0.92	0.5785	0.6764	0.8552	1.0056
0.32	0.9315	0.9506	0.9799	1.9219	0.94	0.5658	0.6658	0.8498	1.0031
0.34	0.9231	0.9445	0.9774	1.8229	0.96	0.5532	0.6551	0.8444	1.0014
0.36	0.9143	0.938	0.9747	1.7358	0.98	0.5407	0.6445	0.8389	1.0003
0.38	0.9052	0.9313	0.9719	1.6587	1	0.5283	0.6339	0.8333	1
0.4	0.8956	0.9243	0.969	1.5901	1.02	0.516	0.6234	0.8278	1.0003
0.42	0.8857	0.917	0.9659	1.5289	1.04	0.5039	0.6129	0.8222	1.0013
0.44	0.8755	0.9094	0.9627	1.474	1.06	0.4919	0.6024	0.8165	1.0029
0.46	0.865	0.9016	0.9594	1.4246	1.08	0.48	0.592	0.8108	1.0051
0.48	0.8541	0.8935	0.9559	1.3801	1.1	0.4684	0.5817	0.8052	1.0079
0.5	0.843	0.8852	0.9524	1.3398	1.12	0.4568	0.5714	0.7994	1.0113
0.52	0.8317	0.8766	0.9487	1.3034	1.14	0.4455	0.5612	0.7937	1.0153
0.54	0.8201	0.8679	0.9449	1.2703	1.16	0.4343	0.5511	0.7879	1.0198
0.56	0.8082	0.8589	0.941	1.2403	1.18	0.4232	0.5411	0.7822	1.0248
0.58	0.7962	0.8498	0.937	1.213	1.2	0.4124	0.5311	0.7764	1.0304
0.6	0.784	0.8405	0.9328	1.1882	1.22	0.4017	0.5213	0.7706	1.0366
TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)-cont'd
M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*
1.24	0.3912	0.5115	0.7648	1.0432	1.88	0.1539	0.2627	0.5859	1.5308
1.26	0.3809	0.5019	0.759	1.0504	1.9	0.1492	0.257	0.5807	1.5553
1.28	0.3708	0.4923	0.7532	1.0581	1.92	0.1447	0.2514	0.5756	1.5804
1.3	0.3609	0.4829	0.7474	1.0663	1.94	0.1403	0.2459	0.5705	1.6062
1.32	0.3512	0.4736	0.7416	1.075	1.96	0.136	0.2405	0.5655	1.6326
1.34	0.3417	0.4644	0.7358	1.0842	1.98	0.1318	0.2352	0.5605	1.6597
1.36	0.3323	0.4553	0.73	1.094	2	0.1278	0.23	0.5556	1.6875
1.38	0.3232	0.4463	0.7242	1.1042	2.02	0.1239	0.225	0.5506	1.716
1.4	0.3142	0.4374	0.7184	1.1149	2.04	0.1201	0.22	0.5458	1.7451
1.42	0.3055	0.4287	0.7126	1.1262	2.06	0.1164	0.2152	0.5409	1.775
1.44	0.2969	0.4201	0.7069	1.1379	2.08	0.1128	0.2104	0.5361	1.8056
1.46	0.2886	0.4116	0.7011	1.1501	2.1	0.1094	0.2058	0.5313	1.8369
1.48	0.2804	0.4032	0.6954	1.1629	2.12	0.106	0.2013	0.5266	1.869
1.5	0.2724	0.395	0.6897	1.1762	2.14	0.1027	0.1968	0.5219	1.9018
1.52	0.2646	0.3869	0.684	1.1899	2.16	0.0996	0.1925	0.5173	1.9354
1.54	0.257	0.3789	0.6783	1.2042	2.18	0.0965	0.1882	0.5127	1.9698
1.56	0.2496	0.371	0.6726	1.219	2.2	0.0935	0.1841	0.5081	2.005
1.58	0.2423	0.3633	0.667	1.2344	2.22	0.0906	0.18	0.5036	2.0409
1.6	0.2353	0.3557	0.6614	1.2502	2.24	0.0878	0.176	0.4991	2.0777
1.62	0.2284	0.3483	0.6558	1.2666	2.26	0.0851	0.1721	0.4947	2.1153
1.64	0.2217	0.3409	0.6502	1.2836	2.28	0.0825	0.1683	0.4903	2.1538
1.66	0.2151	0.3337	0.6447	1.301	2.3	0.08	0.1646	0.4859	2.1931
1.68	0.2088	0.3266	0.6392	1.319	2.32	0.0775	1.1609	0.4816	2.2333
1.7	0.2026	0.3197	0.6337	1.3376	2.34	0.0751	0.1574	0.4773	2.2744
1.72	0.1966	0.3129	0.6283	1.3567	2.36	0.0728	0.1539	0.4731	2.3164
1.74	0.1907	0.3062	0.6229	1.3764	2.38	0.0706	0.1505	0.4688	2.3593
1.76	0.185	0.2996	0.6175	1.3967	2.4	0.0684	0.1472	0.4647	2.4031
1.78	0.1794	0.2931	0.6121	1.4175	2.42	0.0663	0.1439	0.4606	2.4479
1.8	0.174	0.2868	0.6068	1.439	2.44	0.0643	0.1408	0.4565	2.4936
1.82	0.1688	0.2806	0.6015	1.461	2.46	0.0623	0.1377	0.4524	2.5403
1.84	0.1637	0.2745	0.5963	1.4836	2.48	0.0604	0.1346	0.4484	2.588
1.86	0.1587	0.2686	0.591	1.5069	2.5	0.0585	0.1317	0.4444	2.6367
TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)-cont'd
M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*
2.52	0.0567	0.1288	0.4405	2.6865	3.16	0.0215	0.0643	0.3337	4.9304
2.54	0.055	0.126	0.4366	2.7372	3.18	0.0208	0.063	0.3309	5.0248
2.56	0.0533	0.1232	0.4328	2.7891	3.2	0.0202	0.0617	0.3281	5.121
2.58	0.0517	0.1205	0.4289	2.842	3.22	0.0196	0.0604	0.3253	5.2189
2.6	0.0501	0.1179	0.4252	2.896	3.24	0.0191	0.0591	0.3226	5.3186
2.62	0.0486	0.1153	0.4214	2.9511	3.26	0.0185	0.0579	0.3199	5.4201
2.64	0.0471	0.1128	0.4177	3.0073	3.28	0.018	0.0567	0.3173	5.5234
2.66	0.0457	0.1103	0.4141	3.0647	3.3	0.0175	0.0555	0.3147	5.6286
2.68	0.0443	0.1079	0.4104	3.1233	3.32	0.017	0.0544	0.3121	5.7358
2.7	0.043	0.1056	0.4068	3.183	3.34	0.0165	0.0533	0.3095	5.8448
2.72	0.0417	0.1033	0.4033	3.244	3.36	0.016	0.0522	0.3069	5.9558
2.74	0.0404	0.101	0.3998	3.3061	3.38	0.0156	0.0511	0.3044	6.0687
2.76	0.0392	0.0989	0.3963	3.3695	3.4	0.0151	0.0501	0.3019	6.1837
2.78	0.038	0.0967	0.3928	3.4342	3.42	0.0147	0.0491	0.2995	6.3007
2.8	0.0368	0.0946	0.3894	3.5001	3.44	0.0143	0.0481	0.297	6.4198
2.82	0.0357	0.0926	0.386	3.5674	3.46	0.0139	0.0471	0.2946	6.5409
2.84	0.0347	0.0906	0.3827	3.6359	3.48	0.0135	0.0462	0.2922	6.6642
2.86	0.0336	0.0886	0.3794	3.7058	3.5	0.0131	0.0452	0.2899	6.7896
2.88	0.0326	0.0867	0.3761	3.7771	3.52	0.0127	0.0443	0.2875	6.9172
2.9	0.0317	0.0849	0.3729	3.8498	3.54	0.0124	0.0434	0.2852	7.0471
2.92	0.0307	0.0831	0.3696	3.9238	3.56	0.012	0.0426	0.2829	7.1791
2.94	0.0298	0.0813	0.3665	3.9993	3.58	0.0117	0.0417	0.2806	7.3135
2.96	0.0289	0.0796	0.3633	4.0763	3.6	0.0114	0.0409	0.2784	7.4501
2.98	0.0281	0.0779	0.3602	4.1547	3.62	0.0111	0.0401	0.2762	7.5891
3	0.0272	0.0762	0.3571	4.2346	3.64	0.0108	0.0393	0.274	7.7305
3.02	0.0264	0.0746	0.3541	4.316	3.66	0.0105	0.0385	0.2718	7.8742
3.04	0.0256	0.073	0.3511	4.399	3.68	0.0102	0.0378	0.2697	8.0204
3.06	0.0249	0.0715	0.3481	4.4835	3.7	0.0099	0.037	0.2675	8.1691
3.08	0.0242	0.07	0.3452	4.5696	3.72	0.0096	0.0363	0.2654	8.3202
3.1	0.0234	0.0685	0.3422	4.6573	3.74	0.0094	0.0356	0.2633	8.4739
3.12	0.0228	0.0671	0.3393	4.7467	3.76	0.0091	0.0349	0.2613	8.6302
3.14	0.0221	0.0657	0.3365	4.8377	3.78	0.0089	0.0342	0.2592	8.7891
TABLE 15.1 Isentropic Flow of a Perfect Gas (γ = 1.4)-cont'd
M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*	M	[latex]p/p_0[/latex]	[latex] ho/ ho_0[/latex]	[latex]T/T_0[/latex]	A/A*
3.8	0.0086	0.0335	0.2572	8.9506	4.42	0.0038	0.0187	0.2038	15.4724
3.82	0.0084	0.0329	0.2552	9.1148	4.44	0.0037	0.0184	0.2023	15.7388
3.84	0.0082	0.0323	0.2532	0.2817	4.46	0.0036	0.0181	0.2009	16.0092
3.86	0.008	0.0316	0.2513	9.4513	4.48	0.0035	0.0178	0.1994	16.2837
3.88	0.0077	0.031	0.2493	9.6237	4.5	0.0035	0.0174	0.198	16.5622
3.9	0.0075	0.0304	0.2474	9.799	4.52	0.0034	0.0171	0.1966	16.8449
3.92	0.0073	0.0299	0.2455	9.9771	4.54	0.0033	0.0168	0.1952	17.1317
3.94	0.0071	0.0293	0.2436	10.1581	4.56	0.0032	0.0165	0.1938	17.4228
3.96	0.0069	0.0287	0.2418	10.342	4.58	0.0031	0.0163	0.1925	17.7181
3.98	0.0068	0.0282	0.2399	10.5289	4.6	0.0031	0.016	0.1911	18.0178
4	0.0066	0.0277	0.2381	10.7188	4.62	0.003	0.0157	0.1898	18.3218
4.02	0.0064	0.0271	0.2363	10.9117	4.64	0.0029	0.0154	0.1885	18.6303
4.04	0.0062	0.0266	0.2345	11.1077	4.66	0.0028	0.0152	0.1872	18.9433
4.06	0.0061	0.0261	0.2327	11.3068	4.68	0.0028	0.0149	0.1859	19.2608
4.08	0.0059	0.0256	0.231	11.5091	4.7	0.0027	0.0146	0.1846	19.5828
4.1	0.0058	0.0252	0.2293	11.7147	4.72	0.0026	0.0144	0.1833	19.9095
4.12	0.0056	0.0247	0.2275	11.9234	4.74	0.0026	0.0141	0.182	20.2409
4.14	0.0055	0.0242	0.2258	12.1354	4.76	0.0025	0.0139	0.1808	20.577
4.16	0.0053	0.0238	0.2242	12.3508	4.78	0.0025	0.0137	0.1795	20.9179
4.18	0.0052	0.0234	0.2225	12.5695	4.8	0.0024	0.0134	0.1783	21.2637
4.2	0.0051	0.0229	0.2208	12.7916	4.82	0.0023	0.0132	0.1771	21.6144
4.22	0.0049	0.0225	0.2192	13.0172	4.84	0.0023	0.013	0.1759	21.97
4.24	0.0048	0.0221	0.2176	13.2463	4.86	0.0022	0.0128	0.1747	22.3306
4.26	0.0047	0.0217	0.216	13.4789	4.88	0.0022	0.0125	0.1735	22.6963
4.28	0.0046	0.0213	0.2144	13.7151	4.9	0.0021	0.0123	0.1724	23.0671
4.3	0.0044	0.0209	0.2129	13.9549	4.92	0.0021	0.0121	0.1712	23.4431
4.32	0.0043	0.0205	0.2113	14.1984	4.94	0.002	0.0119	0.17	23.8243
4.34	0.0042	0.0202	0.2098	14.4456	4.96	0.002	0.0117	0.1689	24.2109
4.36	0.0041	0.0198	0.2083	14.6965	4.98	0.0019	0.0115	0.1678	24.6027
4.38	0.004	0.0194	0.2067	14.9513	5	0.0019	0.0113	0.1667	25
4.4	0.0039	0.0191	0.2053	15.2099

[latex]p_2=\frac{p_2}{p_0} \frac{p_0}{p_1} p_1=0.1051 \frac{1}{0.0272}(1.0 atm )=3.86 atm[/latex].,

Here, both [latex]M_2[/latex] and [latex]p_2[/latex] are larger than those for (a) because this flow is isentropic while the oblique shock in (a) is not.
For (c), the Prandtl-Meyer function may again be used. Here again the initial flow angle is 0° and [latex] u\left(M_1=3 ight)=49.76^{\circ} ext {. Thus, } u\left(M_2 ight)=0+20^{\circ}+49.76=69.76^{\circ} ext {, for which } M_2=4.31[/latex]. The downstream pressure can be recovered from the Table 15.1:

[latex]p_2=\frac{p_2}{p_0} \frac{p_0}{p_1} p_1=0.0044 \frac{1}{0.0272}(1.0 atm )=0.162 atm[/latex]

Question 15.8: A uniform flow at atmospheric pressure having M1 = 3.0 is de...

Related Answered Questions

Verified Answer:

A normal shock wave forms just ahead of a bullet as it travels at 750 m/s through still air at 100 kPa and 295 K. What are the pressure, temperature, and density of the air immediately behind the shock wave? ...

Verified Answer:

Verified Answer:

Determine the lift and drag coefficients of an infinitely-thin but mildly-cambered airfoil at zero angle of attack in a horizontal supersonic flow at speed M∞. ...

Verified Answer:

Verified Answer:

Consider a quiescent inviscid perfect gas to the right of an ideal piston in a long duct with constant cross-section. At t = t1, the piston located at x = cot1 accelerates to the left until it reaches x = 0, then its speed remains constant. The piston’s trajectory, xp(t), is such that the C+ ...

Verified Answer:

Verified Answer:

Verified Answer: