A Discrete-Ordinates Solution for the Strong Evaporation Problem in Rarefied Gas Dynamics

SCHERER, C. S.

doi:10.5540/tcam.2021.022.02.00201

C. S. SCHERER

Escola Politécnica, Universidade do Vale do Rio do Sinos - UNISINOS, Av. Unisinos, 950, Cristo Rei 93022-750, São Leopoldo, RS, Brasil - E-mail: csscherer@unisinos.br

https://orcid.org/0000-0003-2516-555X

Tables (6)
Formulas (130)

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Table 1:
Ratio

n_{\infty} / n_{0}

.

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Table 2:
Ratio

T_{\infty} / T_{0}

.

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Table 3:
Density perturbation ∆N(τ).

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Table 4:
Velocity perturbation ∆U(τ).

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Table 5:
Temperature perturbation ∆T(τ).

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Table 6:
Heat flux Q _x (τ).

(2.1)

(2.2)

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11 a)

(2.11 b)

(2.11 c)

(2.12)

(2.13)

(2.14)

(2.15 a)

(2.15 b)

(2.15 c)

(2.16)

(2.17)

(2.18)

(2.19)

(2.20)

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

(3.7)

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

(3.13)

(3.14)

(3.15)

(3.16)

(3.17)

(3.18)

(3.19)

(3.20)

(3.21)

(3.22)

(3.23)

(3.24)

(3.25)

(3.26)

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

(4.7 a)

(4.7 b)

(4.7 c)

(4.7 d)

(4.8)

(4.9)

(4.10)

(4.11)

(4.12)

(4.13 a)

(4.13 b)

(4.13 c)

(4.13 d)

(4.14 a)

(4.14 b)

(4.14 c)

(4.14 d)

(4.15)

(4.16)

(4.17)

(4.18)

(4.19)

(4.20)

(4.21)

(4.22)

(4.23)

(4.24)

(4.25)

(4.26)

(4.27)

(4.28)

(4.29)

(4.30)

(4.31)

(4.32)

(4.33)

(4.34)

(4.35)

(4.36)

(4.37)

(4.38)

(4.39)

(4.40)

(4.41 a)

(4.41 b)

(4.41 c)

(4.41 d)

(4.41 e)

(4.42)

(4.43)

(4.44)

(4.45)

(4.46)

(4.47)

(4.48)

(4.49)

(4.50)

(4.51)

(4.52)

(4.53)

(6.1)

(6.2)

	Previous work ¹³13. C.S. Scherer . An Analytical Approach to the Strong Evaporation Problem in Rarefied Gas Dynamics. Z. Angew. Math. Phys., 66 (2015), 1821-1833.		Present work		Ytrehus ²⁵25. T. Ytrehus . Theory and Experiments on Gas Kinetics in Evaporation. In “10th International Symposium on Rarefied Gas Dynamics”. Aspen (1976), pp. 1197-1212.
u	Linear	PP	Linear	PP
0.0	1.00000	1.00000	1.00000	1.00000	1.000
0.1	8.81170(-1)	8.73324(-1)	8.61415(-1)	8.48948(-1)	8.494(-1)
0.2	7.96123(-1)	7.72378(-1)	7.65173(-1)	7.27601(-1)	7.283(-1)
0.3	7.32184(-1)	6.90327(-1)	6.94942(-1)	6.29386(-1)	6.303(-1)
0.4	6.82430(-1)	6.22488(-1)	6.41788(-1)	5.49278(-1)	5.501(-1)
0.5	6.42727(-1)	5.65542(-1)	6.00448(-1)	4.83468(-1)	4.841(-1)
0.6	6.10439(-1)	5.17090(-1)	5.67628(-1)	4.29068(-1)	4.292(-1)
0.7	5.83801(-1)	4.75381(-1)	5.41176(-1)	3.83870(-1)	3.834(-1)
0.8	5.61584(-1)	4.39135(-1)	5.19638(-1)	3.46183(-1)	3.447(-1)
0.9	5.42908(-1)	4.07414(-1)	5.02020(-1)	3.14713(-1)	3.120(-1)

	Previous work ¹³13. C.S. Scherer . An Analytical Approach to the Strong Evaporation Problem in Rarefied Gas Dynamics. Z. Angew. Math. Phys., 66 (2015), 1821-1833.		Present work		Ytrehus ²⁵25. T. Ytrehus . Theory and Experiments on Gas Kinetics in Evaporation. In “10th International Symposium on Rarefied Gas Dynamics”. Aspen (1976), pp. 1197-1212.
u	Linear	PP	Linear	PP
0.0	1.00000	1.00000	1.00000	1.00000	1.000
0.1	9.19458(-1)	9.20433(-1)	9.55222(-1)	9.56753(-1)	9.567(-1)
0.2	8.42106(-1)	8.45652(-1)	9.10103(-1)	9.15865(-1)	9.152(-1)
0.3	7.69530(-1)	7.76876(-1)	8.64777(-1)	8.76913(-1)	8.756(-1)
0.4	7.02368(-1)	7.14499(-1)	6.41788(-1)	8.39659(-1)	8.378(-1)
0.5	6.40748(-1)	6.58453(-1)	7.74781(-1)	8.03915(-1)	8.016(-1)
0.6	5.84522(-1)	6.08403(-1)	7.30756(-1)	7.69501(-1)	7.671(-1)
0.7	5.33391(-1)	5.63854(-1)	6.87722(-1)	7.36245(-1)	7.342(-1)
0.8	4.86987(-1)	5.24234(-1)	6.45835(-1)	7.03974(-1)	7.028(-1)
0.9	4.44912(-1)	4.88933(-1)	6.05152(-1)	6.72524(-1)	6.729(-1)

	$u = 0.5$		$u = 0.8$		$u = 0.9$
τ	Linear	PP	Linear	PP	Linear	PP
0.0	3.74170(-1)	4.078(-1)	8.08298(-1)	1.070	9.77886(-1)	1.521
0.1	3.10046(-1)	3.216(-1)	7.69429(-1)	9.258(-1)	9.72630(-1)	1.361
0.2	2.72970(-1)	2.763(-1)	7.42591(-1)	8.459(-1)	9.68785(-1)	1.277
0.3	2.44505(-1)	2.433(-1)	7.19507(-1)	7.869(-1)	9.65348(-1)	1.216
0.4	2.21154(-1)	2.174(-1)	6.98666(-1)	7.399(-1)	9.62139(-1)	1.170
0.5	2.01347(-1)	1.961(-1)	6.79404(-1)	7.008(-1)	9.59081(-1)	1.133
0.6	1.84195(-1)	1.782(-1)	6.61356(-1)	6.675(-1)	9.56131(-1)	1.102
0.7	1.69133(-1)	1.627(-1)	6.44292(-1)	6.384(-1)	9.53265(-1)	1.076
0.8	1.55770(-1)	1.492(-1)	6.28057(-1)	6.128(-1)	9.50465(-1)	1.054
0.9	1.43819(-1)	1.373(-1)	6.12538(-1)	5.898(-1)	9.47720(-1)	1.035
1.0	1.33065(-1)	1.268(-1)	5.97650(-1)	5.690(-1)	9.45020(-1)	1.018
2.0	6.57847(-2)	6.254(-2)	4.73550(-1)	4.274(-1)	9.19560(-1)	9.226(-1)
5.0	1.10003(-2)	1.073(-2)	2.47078(-1)	2.256(-1)	8.50492(-1)	8.224(-1)
7.0	3.74891(-3)	3.691(-3)	1.61927(-1)	1.513(-1)	8.07905(-1)	7.802(-1)

	$u = 0.5$		$u = 0.8$		$u = 0.9$
τ	Linear	PP	Linear	PP	Linear	PP
0.0	-3.74170(-1)	-2.897(-1)	-8.08298(-1)	-5.170(-1)	-9.77886(-1)	-6.034(-1)
0.1	-3.10046(-1)	-2.433(-1)	-7.69429(-1)	-4.807(-1)	-9.72630(-1)	-5.766(-1)
0.2	-2.72970(-1)	-2.164(-1)	-7.42591(-1)	-4.582(-1)	-9.68785(-1)	-5.608(-1)
0.3	-2.44505(-1)	-1.957(-1)	-7.19507(-1)	-4.403(-1)	-9.65348(-1)	-5.489(-1)
0.4	-2.21154(-1)	-1.786(-1)	-6.98666(-1)	-4.252(-1)	-9.62139(-1)	-5.392(-1)
0.5	-2.01347(-1)	-1.640(-1)	-6.79404(-1)	-4.120(-1)	-9.59081(-1)	-5.311(-1)
0.6	-1.84195(-1)	-1.512(-1)	-6.61356(-1)	-4.003(-1)	-9.56131(-1)	-5.243(-1)
0.7	-1.69133(-1)	-1.399(-1)	-6.44292(-1)	-3.896(-1)	-9.53265(-1)	-5.183(-1)
0.8	-1.55770(-1)	-1.298(-1)	-6.28057(-1)	-3.799(-1)	-9.50465(-1)	-5.132(-1)
0.9	-1.43819(-1)	-1.207(-1)	-6.12538(-1)	-3.709(-1)	-9.47720(-1)	-5.086(-1)
1.0	-1.33065(-1)	-1.125(-1)	-5.97650(-1)	-3.626(-1)	-9.45020(-1)	-5.046(-1)
2.0	-6.57847(-2)	-5.885(-2)	-4.73550(-1)	-2.994(-1)	-9.19560(-1)	-4.798(-1)
5.0	-1.10003(-2)	-1.062(-2)	-2.47078(-1)	-1.840(-1)	-8.50492(-1)	-4.512(-1)
7.0	-3.74891(-3)	-3.678(-3)	-1.61927(-1)	-1.314(-1)	-8.07905(-1)	-4.382(-1)

	$u = 0.5$		$u = 0.8$		$u = 0.9$
τ	Linear	PP	Linear	PP	Linear	PP
0.0	9.63884(-2)	7.013(-2)	4.27184(-1)	2.066(-1)	6.35434(-1)	2.861(-1)
0.1	8.48578(-2)	6.418(-2)	4.10018(-1)	2.156(-1)	6.32504(-1)	3.092(-1)
0.2	7.66260(-2)	5.927(-2)	3.97059(-1)	2.188(-1)	6.30202(-1)	3.221(-1)
0.3	6.98330(-2)	5.491(-2)	3.85566(-1)	2.200(-1)	6.28092(-1)	3.315(-1)
0.4	6.40061(-2)	5.098(-2)	3.75001(-1)	2.200(-1)	6.26094(-1)	3.387(-1)
0.5	5.89040(-2)	4.740(-2)	3.65116(-1)	2.191(-1)	6.24172(-1)	3.445(-1)
0.6	5.43773(-2)	4.413(-2)	3.55771(-1)	2.177(-1)	6.22306(-1)	3.492(-1)
0.7	5.03237(-2)	4.113(-2)	3.46876(-1)	2.159(-1)	6.20484(-1)	3.530(-1)
0.8	4.66683(-2)	3.837(-2)	3.38367(-1)	2.137(-1)	6.18697(-1)	3.561(-1)
0.9	4.33540(-2)	3.584(-2)	3.30199(-1)	2.113(-1)	6.16939(-1)	3.587(-1)
1.0	4.03359(-2)	3.349(-2)	3.22335(-1)	2.088(-1)	6.15206(-1)	3.609(-1)
2.0	2.06982(-2)	1.763(-2)	2.56089(-1)	1.797(-1)	5.98749(-1)	3.685(-1)
5.0	3.62361(-3)	3.154(-3)	1.33810(-1)	1.069(-1)	5.53819(-1)	3.534(-1)
7.0	1.24777(-3)	1.093(-3)	8.76890(-2)	7.444(-2)	5.26085(-1)	3.392(-1)

	$u = 0.5$		$u = 0.8$		$u = 0.9$
τ	Linear	PP	Linear	PP	Linear	PP
0.0	1.14793(-1)	1.027(-1)	1.34017(-1)	3.025(-1)	2.22613(-2)	4.221(-1)
0.1	9.13798(-2)	8.094(-2)	1.23520(-1)	2.441(-1)	2.14856(-2)	3.438(-1)
0.2	7.90159(-2)	6.964(-2)	1.17601(-1)	2.114(-1)	2.11341(-2)	2.992(-1)
0.3	6.98781(-2)	6.147(-2)	1.12925(-1)	1.875(-1)	2.08887(-2)	2.661(-1)
0.4	6.25727(-2)	5.506(-2)	1.08931(-1)	1.687(-1)	2.06976(-2)	2.399(-1)
0.5	5.64955(-2)	4.981(-2)	1.05384(-1)	1.535(-1)	2.05399(-2)	2.186(-1)
0.6	5.13148(-2)	4.538(-2)	1.02159(-1)	1.409(-1)	2.04046(-2)	2.008(-1)
0.7	4.68241(-2)	4.158(-2)	9.91830(-2)	1.303(-1)	2.02854(-2)	1.858(-1)
0.8	4.28838(-2)	3.826(-2)	9.64052(-2)	1.213(-1)	2.01782(-2)	1.730(-1)
0.9	3.93941(-2)	3.533(-2)	9.37919(-2)	1.136(-1)	2.00804(-2)	1.620(-1)
1.0	3.62807(-2)	3.272(-2)	9.13181(-2)	1.068(-1)	1.99899(-2)	1.525(-1)
2.0	1.73687(-2)	1.663(-2)	7.15326(-2)	6.974(-2)	1.92932(-2)	1.022(-1)
5.0	2.78246(-3)	2.958(-3)	3.70904(-2)	3.632(-2)	1.77870(-2)	7.754(-2)
7.0	9.38625(-4)	1.020(-3)	2.43155(-2)	2.466(-2)	1.68996(-2)	7.650(-2)

Brasil

Brasil

A Discrete-Ordinates Solution for the Strong Evaporation Problem in Rarefied Gas Dynamics

ABSTRACT