Improved Exponential Integral Approximation for Tangent-Slab Radiation Transport

摘要

THE tangent-slab approximation is often used to compute shock-nlayer radiation for reentry vehicles [1].With this approximation,nthe radiative flux directed to the vehicle wall is a function of thenproperties along the line of sight normal to the body. Thus, the wall-ndirected radiative flux at point z along the line of sight iswritten as [2]nqu0002nu0001 u0003zu0004u0005 2u0004nZ su0005zsnsu0005znju0001u0003su0004nu0001nu0001nu0001nu0001ndE3u0006u0003u0003s; zu0004u0007ndu0003nu0001nu0001nu0001nu0001nds (1)nwhere zs is the outer limit of integration (such as the shock location),nu0003u0003s; zu0004 is the optical thickness defined asnu0003u0003s; zu0004u0005nZ znsnu0002u0001 dz (2)nand ju0001 and u0002u0001 are the emission and absorption coefficients,nrespectively. The function E3 in Eq. (1) represents the third-ordernexponential integral, defined as E3u0003xu0004u0005nZ 1n1n!u00023nexpu0003u0002!xu0004 d! (3)nfor which there is no exact analytic solution.nThe numerical solution of Eqs. (1–3) is required at each spatial andnspectral point in the shock layer, which for typical cases consists ofnmore than 1 u0001 106npoints. As a result of this large number of points,nan efficient solution procedure is desired for these equations. Ancommon approach [3,4] is to approximate E3 with an exponentialnfunction of the following form:nE3u0003xu0004u0005 1n2neu0002axn(4)nwhere a is a constant. Values for a have been cited by numerousnresearchers: Hunt and Sibulkin (HS) [5], a u0005 2:0; Siegel and Howelln(SH) [6], a u0005 1:8;Ozisik (OZ) [7], a u0005 1:562; andModest (MO) [8],na u0005 1:5. Another approximation was presented by Murty (MU) [9]nin a slightly different form:nE3u0003xu0004u0005 0:30eu00021:1613xnb 0:222eu00022:941xn(5)nThe exponential form of these approximations allows for simpli-nfications in the numerical evaluation of Eqs. (1–3). This is seen bynsubstituting Eq. (4) into Eq. (1), resulting innqu0002nu0001 u0003zu0004u0005 u0004aeu0002au0003u00030;zu0004nZ su0005zsnsu0005znju0001u0003su0004eau0003u00030;su0004nds (6)nThis form is advantageous because the integrand is not a function ofnz, which significantly simplifies its numerical evaluation. Annanalogous formof Eq. (6)may also bewritten using Eq. (5).Note thatnother proposed nonexponential E3 approximations [10–12] do notnallow for this simplification.nFigure 1 compares the exact values of E3 with the HS [5], SH [6],nand MU [9] approximations. A noticeable disagreement with thenexact function is seen for the HS and SH approximations, while thenMU approximation agrees relatively well. Although not shown innFig. 1, similar disagreement is obtained using the OZ [7] andMO [8]napproximations. The influence of this disagreement on shock-layernradiative flux predictions may be assessed by assuming the shocknlayer is a constant property layer.With this assumption, Eq. (1) maynbe integrated analytically to obtain the following:nqu0002nu0001 u0003zu0004u0005 2u0004nju0001nu0002u0001nu0006E3u00030u0004u0002 E3u0003xu0004u0007 (7)nwhere x u0005 u0003u00030;zu0004, which is equal to u0002u0001z (and z is the thickness of thenlayer). By studying Eq. (7), the qualities of an E3 approximation thatnwill lead to an accurate prediction of shock-layer radiative flux maynbe obtained. For instance, observing that E3u00030u0004u0005 0:5 andnE3u00031:5u0004u0005 0:0567, it is concluded that the accuracy of E3 for x 1:5. This is true because thencontribution of E3u0003xu0004 to the bracketed difference in Eq. (7) is smallnfor x> 1:5. Next, note that for optically thin conditions (x t 1),nEq. (7) may be expanded in a Taylor series asnqu0002nu0001 u0003x t 1u0004u0005u00022u0004nju0001nu0002u0001nxndE3u00030u0004ndxnb Ou0003x2nu0004 (8)nThis shows that the derivative of E3 at x u0005 0 must be accurate for thenE3 approximation to accurately model optically thin conditions.nLastly, for optically thick conditions (x n 1), Eq. (7) reduces tonqu0002nu0001 u0003zu0004u0005 2u0004nju0001nu0002u0001nE3u00030u0004b Ou0003eu0002xn=xu0004 (9) nwhich implies that the value of E3 at x u0005 0 must be accurate for thenapproximation to accurately model optically thick conditions.nThe previous paragraph suggests the following three criteria forndeveloping an E3 approximation that provides an accurate predictionnof shock-layer radiative flux:n1) The accuracy of E3 for x< 1:5 ismore important than for largernx values.n2) The E3 derivative at x u0005 0 must be accurately modeled.n3) The E3 value at x u0005 0 must be accurately modeled.nOf the previous approximations mentioned in the discussion ofnEqs. (4) and (5), all except theMU approximation satisfy criterion 2,nwhereas only the HS and MU approximations satisfy criterion 3.nRegarding criterion 1, it will be shown in the next section that allnmentioned approximations, except the MU approximation, havenmaximum errors greater than 5%for x values below 1.5. The subjectnof the remainder of this Note is to find an approximation that has annerror of less than 5% for values of x below 1.5 while also satisfyingncriteria 2 and 3.