Any point on the neutral surface has a zero eigenvalue by definition and thus satisfies the following conditions which are obtained by setting A = 0 in Eq.( 2.51).
IP and IP' are the right and left eigenvectors of L0 respectively and L'a is the adjoint operator defined by the inner product relation for general vectors W and <P
In this case L'0 is the complex conjugate adjoint of the matrix operator La appearing in Eq.( 2.51). For the numerical calculations the discrete inner product is applied such that if A. B € if4N+3 then this relation is defined as follows where an overbar denotes complex conjugation.
In order to determine conditions for the neutral point in (Q./J, R) parameter space con sider the following generalised eigenvalue problem. (Eq.( 2.51).)
where subscripts on L and A represent the partial derivatives of L„ and A with respect to that particular variable.
If a neutral paraboloid of the form shown in Figure 2.6 is assumed then 6R/dU = 0 at the critical point. With < * ‘,K IP > / 0 Eq.( 2.59) implies that An must be zero. Similarly by considering L0 = L0(/3(Q)) the condition \g = 0 is also obtained. Therefore at the critical point of the paraboloid the following three conditions must be satisfied
L0¥ = 0 and L’0* ‘ = 0 (2.54)
< W .L 0* >=< L-0W,tP > (2.55)
47V+2
< A ,B >= ^ 2 A¡B¡ (2.56)
L09 = A K * . (2.57)
If L0 = L0(R(il)) then differentiating Eq.( 2.57) above with respect to Q gives
(2.58) Taking the inner product of Eq.( 2.58) with the left eigenvector, IP*, leaves
f) R
^ < * \ L n * >= An < * \ K * > (2.59)
Figure 2.6: Neutral paraboloid in (Í2,0,R) parameter space.
The eigenvalue, A, is now treated as a function of Q, 0 and R and expanded in a Taylor series about the critical point (fic, 0C, Rc). If A0 is the minimum value on the neutral surface then
A — A0 -f- An(if — fie) + A^(/? ~ 0c) + Ah(R — Rc) -+■
—Ann(ii — iic)2 + — 0c)2 + An^fil — &c)(0 — 0c) + • • • (2.61) Expressions for the first and second partial derivatives of A are thus required in order to implement a form of Newton’s method.
Consider the expression
LF = \K F (2.62)
where F is a general eigenvector of L and let A represent any one of the parameters, Q, 0 or R. Differentiating Eq.( 2.62) with respect to A' gives
Lx F + LF X = A x K F + \ K F X ■ (2.63)
Evaluation at the critical point, A = 0, and taking the inner product with the left eigenvector, F '. returns an expression for Ax-
Ax < F \ K F > but < F*,LFx > => Ax < F*,LXF > + < F*, L F X > < L*F",Fx >= 0 by definition < F \ L XF > < F \ K F > (2.64)
variable Y say, which again represents any one of the three parameters involved.
^ { L x F + L F x = XxK F + X K Fx}
Lx yF + Lx Fy -+■ LyFx + LF xy = X xyK F + X xK F y +
XyKFx + \ K Fxy (2.65)
Evaluating at the critical point leaves
LF xy — Xxy K F + X xK F y + XyKF x — Lx yF — LxF y — Ly F x (2.66)
To obtain an expression for the second derivative, Xxy > the following solvability condition
is applied :
Solvability condition.
Suppose F" is a solution of the adjoint problem L’F" = 0 then LG = g is solvable if and only if < F",g >= 0.
In order to prove this the inner product of F" with LG = g is taken. < F ",L G > = < F ‘,g>
but < F", LG > = < L"F", G > = 0 since L ’F" = 0
=> < F",g > = 0 (2.67)
Applying this solvability condition to the expression in Eq.( 2.66) then gives the result A*y < F ’,K F > +A* < F ",K F V > +Ay < F ",K F X > - ao,
(2.68) < F", Lx yF > — < F",LxFy > — < F",LyFx > = 0
or
Xxy = {< F", Lx yF > + < F ', Lx Fy > —Ax < F “, K Fy > —
Ay < F", K F X > }/< F ",K F > . (2.69) The only undetermined values are those of Fx (or F y ) which can be obtained from Eq.( 2.63) where evaluation at the critical point leaves
L0FX = Xx K F - LXF. (2.70)
Once Ax has been calculated it would appear that this could be solved directly through the application of a standard numerical routine but problems arise since L 0 is in fact singular
and Fx must be determined by solving an augmented problem. Eq.( 2.70) is therefore en larged as follows with the details of the method described in § 2.12.
L„
p . T F X
----
1
1 ___i
^X 0 (2.71)
This now gives a non-singular problem from which F \ can be obtained using a standard linear system solver.
In order to apply Newton’s method to this problem an initial estimate (Q, 0, R)0 is made for the critical point, (Qc, 0C, Re). An improvement on this initial value is made by applying the following iterative formula.
where n n A(fi, 0,R)n 0 = 8 - J r ,'1 \ a(n,l3,R)„ R n +1 R n An($l,/J,fl)„ A/s A H ■ Vn -W X/3R •^n/3 -'n r evaluated at (ÎÎ,/?,./?)„ (2.73)
J„ is invertible in the neighbourhood of the critical point since \r ^ 0 and the subdetermi
nant formed from the second derivatives of A with respect to Q and 8 is also non-zero. The process in Eq.( 2.72) is repeated iteratively until some convergence condition is satisfied and in this particular case a convergence tolerance of 10-8 was used.
Throughout the numerical calculations the value of A° is retained to reduce errors in the subsequent calculations although it is assumed to be exactly zero in the analysis.
Numerical values obtained using the above method are shown in Table 2.1. These points correspond to the minima of the neutral paraboloids in (fi, 0, R) parameter space. Values for A, dX/d/3 and 8X/8Q are calculated to be 10-8 or less confirming that the minima have in fact been correctly located. Figures 2.7(a)-(d) show numerically calculated neutral contours for different values of 9 indicating the structure of the neutral paraboloids.
The critical points are found to occur at different rotation rates with higher rotation speeds required to give the minimum as plane Couette flow is approached. The fact that all the states investigated have well defined minimum points suggests that the three-dimensional
neutral paraboloid surface in (i2, /?, R) parameter space exists for the whole family of profiles given by Eqs.( 2.52) and ( 2.53). Existing results for plane Poiseuille flow are in fact very close to the global minimum point calculated here with ii = 0.1667, (3 = 2.45 and R = 66.5 often quoted as the critical parameters. The minimum critical values predicted for Couette flow agree well with those of Lezius and Johnston (1976) given in the review text.
e n c 0c Rc 0 0.1683 2.4592 66.4476 1 3 0.1433 2.2460 51.2680 2 3 0.1718 1.5790 30.5300 i 0.2500 1.5582 20.6625 Table 2.1: Minimum neutral points.
Figure 2.7: Numerically determined neutral contours in (ii, (3) parameter space, (a) : 0 = 0, (b) : 0 = 1, (c) : 9 = (d) : 0= 1. x marks the position of the critical point.