Test Time in a 1.5-Inch-Diameter High-Stagnation-Enthalpy Shock Tube

Abstract

The most popular method of solution of boundary-value problems like the foregoing is founded on Green's function. With its help, the moment on the nth support has the value n = G nk G h (4) To modify this method and to change the finite-difference operator L and boundary values (2) into single operator R or to change the boundary-value problem (1) into the algebraic equationthe operator L will be expanded. The eigenvalues can be found from the equation I -XL = 0 or from the equivalent equationwhere X * cos(kir/N)]and N -1 is the number of rows and columns in matrix L. But it is easy to prove that sin(kirn/N) represents the components of the eigenvectors of L. To carry out the normalization, these quantities must be multiplied by a constant factor, and then one gets the "projection tensors". kirm . kirnThus one can write the operator R, which substitutes for (2) in expanded form, R = P* and for Eq. (5) as follows:From (9), with help of the tensor property of P,one gets, after multiplication from the left by Pz, the explicit solution of the "three-moments equation:"The comparison of (11) with (4) yields Green's function G nm = -\iP nm ;i I = 0,1, ....N (12)The comparison of (12) and (8) reveals that R and G are commuting operators. This property distinguishes the operators R and (2). After summation, it is possible to write Green's function (12) in the form W nk = -|n -k\a--2 sinho-. , , . 7N . 2 sinhncr sinh^cr) , . ffln h(n+t) g + tanh^ \ (13) where sinhcr = 3 1/2 . Then the explicit solution (11) takes more compact form: Mn = (14) Adding to (14) regular part, one can write the solution of the boundary value problem M n+1 + 4M n + M n -i = -G n M 0 = M Q M N = M N n = 0,1, . . . N (15) as follows: M n = WnkG k -(-!)"+» (sinrmoysinhA/V) X [(-1) VTlfo -M N ] + (-l (16)To facilitate insight into the structure of Green's function (12) and (13), one can take, as in the previous case, N = 3, ThenAi = J, X 2 = |,and wm wn G nmFor particular numbers G = -10 + -6> 10 6" ' 1~I ' >10Assuming, for example, N = 3 and the load terms, GI = G 2 = JpZ 2 , one obtains from (15) the following expression foi moments:in agreement with previous results.