Effects of dimensional wall temperature on velocity-temperature correlations in supersonic turbulent channel flow <sc>of thermally perfect gas</sc>

XiaoPing Chen; XinLiang Li; ZuChao Zhu

doi:10.1007/s11433-018-9318-4

SCIENCE CHINA Physics, Mechanics & Astronomy

SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 62, Issue 6: 064711(2019) https://doi.org/10.1007/s11433-018-9318-4

Effects of dimensional wall temperature on velocity-temperature correlations in supersonic turbulent channel flow of thermally perfect gas

XiaoPing Chen¹, XinLiang Li^2,3,*, ZuChao Zhu¹

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ReceivedJul 25, 2018
AcceptedOct 30, 2018
PublishedJan 25, 2019

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Abstract

Direct numerical simulations of temporally evolving supersonic turbulent channel flows of thermally perfect gas are conducted at Mach number 3.0 and Reynolds number 4800 for various values of the dimensional wall temperature to study the influence of the latter on the velocity-temperature correlations. The results show that in a fully developed turbulent channel flow, as the dimensional wall temperature increases, there is little change in the mean velocity, but the mean temperature decreases. The mean temperature is found to be a quadratic function of the mean velocity, the curvature of which increases with increasing dimensional wall temperature. The concept of “recovery enthalpy” provides a connection between the mean velocity and the mean temperature, and is independent of dimensional wall temperature. The right tails of probability density function of the streamwise velocity fluctuation grows with increasing dimensional wall temperature. The dimensional wall temperature does not have a significant influence on the Reynolds analogy factor or strong Reynolds analogy (SRA). The modifications of SRA by Huang et al. and Zhang et al. provide reasonably good results, which are better than those of the modifications by Cebeci and Smith and by Rubesin.

Funded by

the National Natural Science Foundation of China(Grant,Nos.,11502236,51536008,91852203)

the National Key Research and Development Program of China(Grant,No.,2016YFA0401200)

Science Challenge Project(Grant,No.,TZ2016001)

and the Natural Science Foundation of Zhejiang Province(Grant,No.,LQ16E090005)

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11502236, 51536008, and 91852203), the National Key Research and Development Program of China (Grant No. 2016YFA0401200), Science Challenge Project (Grant No. TZ2016001), and the Natural Science Foundation of Zhejiang Province (Grant No. LQ16E090005).

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Figure 1
(Color online) Distribution of vibrational energy excited degree for different dimensional wall temperature.
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Figure 2
(Color online) Distributions of mean velocity (a) and mean temperature (b) for different dimensional wall temperature.
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Figure 3
(Color online) Distribution of van Driest transformed velocity.
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Figure 4
(Color online) Distributions of RMS velocity (a) and RMS temperature fluctuations (b) for different dimensional wall temperature.
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Figure 5
(Color online) Distributions of turbulent Mach number (a) and RMS Mach number fluctuation (b) for different dimensional wall temperature.
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Figure 6
(Color online) PDFs of streamwise velocity fluctuation (a) and temperature fluctuation (b) near the wall (1?|y|=0.04) for different dimensional wall temperature.
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Figure 7
(Color online) PDFs of streamwise velocity fluctuation with the associated sign of temperature fluctuation (a) and temperature fluctuation with the associated sign of streamwise velocity fluctuation (b) near the wall ( $1 ? | y | = 0.04$ ) for different dimensional wall temperature.
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Figure 8
(Color online) Relationship between mean temperature and mean streamwise velocity for WT5 (a) and different dimensional wall temperature (b).
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Figure 9
(Color online) Distribution of “Recovery enthalpy” $h_{r}^{*}$ versus mean streamwise ${u} / u_{e}$ for different dimensional wall temperature.
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Figure 10
(Color online) Distributions of SRA (a), ESRA (b), GSRA (c), RSRA (d), HSRA (e) and GHSRA (f) for different dimensional wall temperature.
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Figure 11
(Color online) (a) Distribution of $R_{u^{″} T^{″}}$ for different dimensional wall temperature; (b) joint probability density of instantaneous streamwise velocity fluctuation versus temperature fluctuation near the wall ( $1 ? | y | = 0.04$ ) for WT5.
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Figure 12
(Color online) Distribution of Pr_t for different dimensional wall temperature.
Table 1 Table 1 Flow and computational parameters for different dimensional wall temperature

Cases
$T_{w}^{*}$ (K)
$M a_{}^{}$
$R e_{}^{}$
$P r_{}^{}$
$T_{w}^{}$
WT1
149.075
3.0
4880
0.7
1.0
WT2
298.15
3.0
4880
0.7
1.0
WT3
596.30
3.0
4880
0.7
1.0
WT4
1192.6
3.0
4880
0.7
1.0
WT5
1788.9
3.0
4880
0.7
1.0

Table 2 Table 2 Grid resolution and domain size for different dimensional wall temperature

Cases	L_x/H	L_y/H	L_z/H	n_x	n_y	n_z	Δx⁺	$Δ y_{w}^{+}$	$Δ y_{\max}^{+}$	Δz⁺
WT1	4π	2	4π/3	571	261	251	10.12	0.243	8.97	7.676
WT2	4π	2	4π/3	571	261	251	9.763	0.230	8.65	7.403
WT3	4π	2	4π/3	571	261	251	9.419	0.226	8.35	7.149
WT4	4π	2	4π/3	571	261	251	9.248	0.222	8.19	7.014
WT5	4π	2	4π/3	571	261	251	9.061	0.218	8.03	6.871

Table 3 Table 3 Time-averaged parameters for different dimensional wall temperature

Cases	Ma_τ	Re_τ	--B_q	$? ρ_{w} ?$	$? ρ_{c}^{} ?$	$? T_{c}^{} ?$	$? μ_{c}^{} ?$	$? γ_{w}^{} ?$	$? γ_{c}^{} ?$
WT1	0.115	460.52	0.1455	2.444	0.948	2.550	2.154	1.400	1.396
WT2	0.118	443.59	0.1344	2.317	0.952	2.380	1.829	1.399	1.366
WT3	0.123	428.25	0.1282	2.142	0.955	2.136	1.598	1.378	1.323
WT4	0.127	420.24	0.1193	2.039	0.957	2.069	1.503	1.325	1.296
WT5	0.128	411.74	0.0952	1.984	0.959	2.029	1.437	1.305	1.291

Table 4 Table 4 Skin friction and heat transfer for different dimensional wall temperature

Cases	C_f (×10^?3)	C_h (×10^?3)	2C_h/C_f	${Q'}_{wrms} / Q_{w}$	${τ'}_{wrms} / τ_{w}$
WT1	7.128	4.401	1.234	0.401	0.389
WT2	7.156	4.294	1.200	0.407	0.385
WT3	7.197	4.247	1.180	0.401	0.391
WT4	7.281	4.208	1.156	0.409	0.399
WT5	7.208	4.199	1.165	0.410	0.402

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47.27.ek	Direct numerical simulations
47.27.nd	Channel flow
47.40.Ki	Supersonic and hypersonic flows

Effects of dimensional wall temperature on velocity-temperature correlations in supersonic turbulent channel flow of thermally perfect gas

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Cases	$T_{w}^{*}$ (K)	$M a_{}^{}$	$R e_{}^{}$	$P r_{}^{}$	$T_{w}^{}$
WT1	149.075	3.0	4880	0.7	1.0
WT2	298.15	3.0	4880	0.7	1.0
WT3	596.30	3.0	4880	0.7	1.0
WT4	1192.6	3.0	4880	0.7	1.0
WT5	1788.9	3.0	4880	0.7	1.0