73

Figure 6.9: Amounts of eigenfunctions (above) and (below) to the eigenvalue with the largest imaginary part in 6.5 for N = 27(----) and N = 39(____). The maximum declines strongly with |n|

Figure 6.10: Estimation of accuracy of the eigenvalues of the group for N = 39.

74

Figure 6.11: Variation of the first eight eigenvalues with the amplitude of the streaks from s = 0:2 to 0:3 .

75

Figure 6.12: Variation of the first eight eigenvalues with the Reynolds-number from Re = 1800 to 6000 .

76

It leads to Re ≈ 3900 in the parameter area under consideration for the unstable value to a positive growth rate i. This means that the modified pipe flow with the final Reynolds-numbers allows increasing linear disturbances, which is not the case with the Hagen-Poiseuille-profile.

Generally the wave number increases with the frequency r, as is clear from figure 6.13. The explanation for the same is that the phase velocity r/ of an eigenfunction orients itself on the local convection velocity, which is independent of . This is in the area of the maxima of the eigenfunction. The eigenvalue of both the branches in the complex levels are stabilized through a higher value of and are opposite for , because the dissipation is at its lowest in this limited case of vanishing gradients in the axial direction. The imaginary portion has a local maxima at ≈ 2.7 through the deformation of the accrued eigenvalue.

6.2.4 Characterization of the eigenfunctions

The least-damped eigenfunctions should be represented below. With each two sections z = z0 and in z = z0 + /4, = 2/, its given wavelength is shown. One gets the velocity vectors at z0 + /2 and z0 + 3/4 obtained from the first two sections, in which the direction of arrow is reversed. From z0+ the cycle begins again with the first image. As a result of this symmetry, the representation is limited to the first quadrant. Figure 6.14 shows the position of the low speed streaks and the high-speed streaks in a schematic manner.

Under the current sections, the levels = 0 (low speed streaks) and = /6 (high speed streaks) are represented in such a way that the eigenfunctions possess the velocity components parallel to these levels. This is given to the group for both the values from . The group

vanishes in the level = 0 whereas the velocities vr and vz are valid for the eigenfunctions from to = /6. Finally these similarities are applied to the eigenfunctions in the levels of low speed streaks and high-speed streaks. The levels constantly summarize a wavelength and are compressed approximately by a factor of 2 in the axial direction. The scale distinguishes itself i.e. from the sections z = constant.

Firstly those eigenfunctions, which are minimally damped, should be considered in the figures 6.15-6.18, whose eigenvalues do not alter themselves practically with the variants of the intensity s of the streaks in the figure 6.11. The frequency r of all these functions nearly achieves the wave number , and thus reaches values of almost 3. The phase velocities r/ nearly approach to the dimensionless velocity from 1 to r = 0.

77

Figure 6.13: Variation of the first eight eigenvalues with the wave number from to 5 .

78

Figure 6.14: Distribution of the high-speed streaks and low speed streaks in the first quadrant

Figure 6.15: Velocities for the eigenvalue ≈ 2.9077 – 0.0921 i of group .

79

Figure 6.16: Velocities for eigenvalue ≈ 2.7709 – 0.1450 i of the group . Under = 0 (levels of low speed streaks)

80

Figure 6.17: Velocities for eigenvalue ≈ 2.9106 – 0.0924 i of group .

From all the above examples, one can thus expect that maxima of the velocities concentrate at the center of the pipe, which is also confirmed by the representations.

Because the streaks from Figure 6.3 deactivate against the center, they hardly interact with these eigenfunctions. The functions in figures 6.15 and 6.17 are dominated by the Fourier-Mode n = 0, which means it hardly diversifies in the azimuthal direction. This is yet another reason as to why it is not influenced by the streaks with n = 6. In the extreme case 0 and ∞, the radial components of the bulging eddy vanish in figure 6.15 and what remains is the reallocation of the momentum either from tube center towards the wall or vice versa. Both the functions mentioned in figures 6.16 and 6.18 and modes n = 3 (2k+1) are influenced by its fundamental mode n = ± 3

The next eigenfunctions from figures 6.19-6.22 belong to the eigenvalues with relatively huge real parts whose damping decreases in contrary to the preceding group with the increasing s. With correspondence to this, the somewhat lower phase velocity lies at a maximum state having higher r. The importance of the contributions of the other modes such as n = 0 and n = ± 3 reduces. The similarities are conspicuous between the eigenfunctions of the adjacent eigenvalues. Hence the Figure 6.15, Figure 6.15 and Figure 6.20 differentiate themselves from the Figure 6.21 in such a way that with the former, azimuthally adjacent eddy turns in the opposite direction, while with the latter all eddies in section with z = const. have the same rotational direction.

Figures 6.23-6.26 exemplify a few eigenfunctions corresponding to the eigenvalues of the left branches in the figure 6.5. The structures concentrate itself near to the wall whereby high gradients are formed together with the no slip condition.

81

Figure 6.18: Velocities for eigenvalues ≈ 2.7624 – 0.1483 i of the group . Below (levels of high speed streaks) = /6.

82

Figure 6.19: Velocities for eigenvalues ≈ 2.7611 – 0.1240 i of group .

83

Figure 6.20: Velocities for eigenvalues ≈ 2.4031 – 0.2443 i of group . The velocities vr

and vz in the = 0 level are very small compared to the axial sections. Therefore they are not represented. On the basis of symmetry vr = vz= 0, it is already in the = /6 level.

Figure 6.21: Velocities for eigenvalues ≈ 2.4045 – 0.2443 i of group .

84

Figure 6.22: Velocities for eigenvalues 2.5407 – 0.0770 i of group . Below (levels of high speed streaks) = /6.

85

Figure 6.23: Velocities for eigenvalues 0.2791 – 0.2483 i of group

86

Figure 6.24: Velocities for eigenvalues 0.3147 – 0.1553 i of group . Below = 0 (levels of the low speed streaks).

87

Figure 6.25: Velocities for eigenvalues 0.5064 – 0.1893 i of group .

The first two velocity fields possess components in the radial direction in the low speed streaks that lead to local momentum exchange over the velocity gradients away from the basic flow (6.1).

Finally in the figures 6.27 and 6.28, both the eigenfunctions should be dealt with, whose eigenvalues in Figure 6.5 lie outside both the main branches. If one compares figure 6.27 to figure 6.24 as well as figure 6.28 to figure 6.25, it becomes clear that no qualitatively new structures exist here. Corresponding to the larger value of r, the maxima shifts itself solely towards the center of the pipe. This is in comparison to the eigenfunctions of the left side of the eigenvalues.

6.2.5 Influence of the profile

The model applied for the streaks is partially arbitrary. In order to be able to estimate the details, a profile should be examined for a comparison that originates directly from the simulation. A similar velocity distribution was selected in Figure 4.14, however, at an axial position z = 4.84 as the process over r is quite smooth and can be represented well as Tschebyscheff-series. For simplification the distribution of the Fourier-modes 0 and 6 were reduced. Both the portions

and are determined in a such a way that the profile of the simulation can be exactly met with at the = k/3 (low speed streaks) and = (2k+1)/6 (high-speed streaks).

88

Figure 6.26: Velocities for eigenvalues 0.3579 – 0.2200i of group . Below = /6 (levels of high speed streaks).

89

Figure 6.27: Velocities for eigenvalues 0.8457 – 0.0142i of group Below = (levels of the low speed streaks).

90

Figure 6.28: Velocities for eigenvalues 1.3542 – 0.1283 i of group

Figure 6.29: Approximation of the medium perturbed temporal velocity profile of the simulation with z = 4.84.

The approximations for and for are seen in the 6.29 and 6.30 figures.

91

Figure 6.30: Superposition of the disturbed velocity profile from Figure 6.29 with the base flow.

The amplitudes of the disturbed velocities move themselves between –0.20 and +0.24 and compare themselves with similar obtained from the linear theory for amplitude of s 0.22. The new accrued axial symmetry component transfers the momentum from the center of the pipe to the wall and reflects the change to turbulence. Hence perturbed components from figure 6.29 have higher gradients in the high-speed streaks at the same amplitude as compared to those in fig. 6.30, whereas the retardation of flow in low speed streaks is less intense.

6.2.6 Relevance for the transition

The question arises, which of the sum of the observed structures leads to the observed laminar-turbulence change. One can then ignore the groups and because they are not activated in the experiment discussed i.e. it cannot occur through the enforced symmetry of the simulation. The remaining eigenfunction from 6.27 figure is especially interesting because the given streaks increase linearly with the Reynolds- numbers. These numbers are more than 4000. The activation described in section 4.1 (P.33) inserts such secondary flows that also represent the early stage of the eddy. The radial velocity in the levels of low speed streaks corresponds to the exchange in momentum through the edges of the - and the - turbulences.

92

If one reverses the question and searches for the dominating structures of transition among the least damped eigenfunctions, the result is disappointing. Besides the streaks, the analysis of the secondary stability is pre-determined as “artificial”, and the change determines the longitudinal eddy just as in figure 4.12. This eddy allocates the momentum of the base flow in such a manner that it forms a roundabout pair from the low speed and the high-speed streaks. Though the eddy structures occur in the linear analysis, as in figure 6.19, nonetheless these lie in the center of the high-speed streaks.

93

7 Coherent structures

The Karhunen-Love-Analysis (proper orthogonal decomposition) (15) represents temporal and spatial alterable flow fields as a series

The functions thus build an orthogonal system. The analysis is optimal in the sense that with higher number of terms in a series (7.1), no other basis could have delivered a better approximation. On the other hand, these basis functions adapt extremely well at a fixed, specified flow and are not represented here for the discretization of any other flow fields for example the spectral procedures.

In the existing case, the specified velocity divisions should be approximated as series at a discrete moment: tm = t0 + m t, m = 0,1,…T. As depicted in Sirovich (23), the basis functions can thus be calculated through the linear superposition

of the given fields. The original data is reconstructed in another way by

The units are described in appendix M (P 154). The control volume of the length with boundary conditions should be examined

periodically in an axial direction as shown in the previous chapter. The spatially shown kinetic energy at the moment tm is again

and is thus set according to

from the energies of the eigenfunctions

One would like to analyze the dynamic behavior of the flow based on the eigenfunction (7.2), thus one can insert the analysis (7.1) in the Navier-Stokes-equations (2.3). A problem arises here that the pressure emanates from the non-linear relation (2.7) of the velocity and does not develop itself in a series like in (7.3).

94

First time step 190 t 600 t 1010 tLast time step 530 t 840 t 1150 tMiddle moment T 2T 3TNumber of eigenfunctions 35 25 15

Table 7.1: Time spacing for the determination of the KL-Eigenfunctions

On the other hand when one carries out the rotation of the momentum conservation equations, the pressure falls. This is because the rotation of the gradients of the scalars basically disappears (3). The following procedure orients itself substantially to that of Rempfer (20). The so-called eddy transport equation is

The Galerkin-projection discussed in appendix N (P 157) then leads to an ordinary, non linear differential equation

The temporal alteration of the kinetic energy is shown as

7.1 Determination of Eigenfunctions

Even here the spatial periodic data should be analyzed again as in the previous chapter. Three time intervals are observed individually, which are listed in table 7.1. The first one is characterized through the formation of the Λ-eddy, the second one through the development of secondary eddy, and the third one through the interaction of the individual eddies with each other.

The images in figures 7.1 and 7.2 represent the strength of eddies of the first ten eigenfunctions in the first time interval. Based on the considerable differences in the amplitudes of the eigenfunctions, the representations are individually scaled i.e. the contour levels lie between the extremes of the respective represented function. The pertinent coefficients τi are clear from the figure 7.3, and the energetic contributions from figure 7.4.

95

Figure 7.1: The intensity of the eddy in the = 0 section of the eigenfunctions 0 to 4 of the first time interval. The base flow is subtracted from the null eigenfunction.

96

Figure 7.2: The intensity of the turbulences in the =0 section of the eigenfunctions 5 to 9 of the first time interval.

97

Figure 7.3: Development of the coefficients τi of the first 10 eigenfunctions during the first time interval. A value 4i was added respectively to the equalization of the representation. discrete values—integrated development equation—trigonometric approach.

98

Figure 7.4: Energetic amount of first five eigenfunctions during the first time interval. Discrete values—integrated development equation.

99

Scope

A Decoupling of momentum conservation equations

One defines the linear operator

The momentum conservation equation in the radial and azimuthal direction is written as

The linear terms that contain the velocity components are on the left-hand side of the equal sign. To the right are the convective terms or the pressure terms.

The aim is to decouple the above-mentioned system, i.e. to find the linear combinations of both the equations

on whose left side only an dependent variable u appears. This is only possible when u1 and u2

are linearly dependent and

the determinants vanish. Thus b = ia must also be valid. If one selects = 1, then one gets both

the parameters as new dependent variables, refer to e.g. (19).

100

B Semi implicit Runge-kutta –behavior

The conservation momentum equations (2.5d) can be derived in the form

Here L is the linear operator from (2.12). N integrates all the remaining terms. In the existing procedure the intermediate sizes are calculated as per Ascher (1) according to

In doing so u0 = u(t0) is the result of the preceding total section at the same point of time t0. The resolution results into

as an identity operator with I.

The parameters are initialized to at the beginning of combination section (total step). Thereby s serves as the number of partial steps. Before the calculation of the new ui, the explicitly integrated terms Ni - 1 are evaluated and are incremented according to

and can be thus calculated from . is then restored to the implicit integrated portions, i.e. . Thus the required terms can be regained from ΔtLui = (ui – vi)/aii. Ignoring the fact that this is efficient than the direct calculation, the operator L is not required. The solution of the combination section (total step) is

The coefficients patterns of the procedure applied, which is described here and in (1) with “’(4.4.3)”’ with four partial stages

101

Thus the coefficients are placed respectively in the left column . They specify the relative time level of the time steps i.e. the boundary values that are dependent on time are evaluated at this moment . The coefficients and stay to its right. In this special illustration, the values and are identical with and in the lowest line. Thus the last sub-result us is similar to the searched value u(t0 + Δt).

A further advantage of this illustration is that the coefficients aii have the same ½ value, so that the operator (I – Δtaii L)-1 must be implemented only once. Besides this the influence matrices differentiate themselves for the partial stages only with constant factors (refer section J (P 119)).

102

C Illustration of exponential series

A scalar function p represents itself as series

If the function represents a physical parameter, it must be analytical. This implies that the derivatives ∂sp / ∂ xs , s > 0 with regards to the Cartesian coordinates x and also for r 0, remain finite. In the following it is represented, that this happens only if specific coefficients are generally set to 0.With the help of the relations

and the partial derivatives

the first derivative computes according to x as

Based on the linearity, summation and the derivative, one can separately observe a single item of (C 1). One can obtain the following through a repeated implementation of (C 2)

This expression remains infinite for s ≤ k and for r 0. However due to either the related coefficient must be for s > k or each single item in (C 3) takes the value as 0.In order to review the latter, it is enough to guarantee this for s = k+1, since all further derivatives s = k +2, k+3…. then disappears, and one obtains the terms

103

First of all, only one of the factors can become 0, if is direct. Hence one puts forth and divides by , and obtains

In the case of i = 0, the second product does not make any contribution. The first includes exactly the factor 0 if one of the j takes the value n + m,

0 n + m n + 2m

or m max (0, -n). (C.4)

For i = n+ 2m +1, analogously one obtains the same condition for the second product from

0 m n + 2m

This is fulfilled from the specific value , thus the first product disappears for all . The second product becomes 0 for . Hence (C 4) is adequate for all i. The coefficients of the series (C 1) must be valid for

and thus

Thus the components of a vector

must remain with the analytical functions. If one transforms the unit vectors and , which are dependent on into the constant Cartesian vectors and , then

For the component vz, the same series representation (C 5) is obtained, as for a scalar. Contrary to this, one obtains items of the form for through the factors. Thus one replaces n by n 1 in the above derivation, hence

104

This is directly followed by

The results obtained correspond to the results given by Orszag and Patera (16).

105

D Recursion relations

A function g (r) can be represented at the interval r (-1, 1) as a final Tschebyscheff series by way of approximation.

The Tschebyscheff polynomials are defined as

The first four polynomials are

T0 = 1, T1 = r, T2 = 2r2- 1, T3 = 4r3 –3r.

Instructions for calculations given in the following sections are derived from base operations. A few can also be found in (7).

D-1 h= rg

A direct calculation specification can be indicated for the coefficients of the product r.g ( r) as per trigonometric identities. One puts forth

One thus obtains from the coefficient comparison

106

The value is ignored. The resolution in the radial direction must hence be adequately large enough so that the spectrum is considerably covered through the final series and the importance of the errors is reduced.

D-2 h = g/r The relations between the coefficients and for a division through r results directly from (D 1), i. e.

The starting values for the recursions are and .If one calculates directly by the addition of the recursion relations (D 2) with alternate

signs for k = 2,4,6… one obtains,

The opposite of (D.1) should be (D.2), thus this value must correspond to the coefficients :

Alternatively one obtains

and the analytical part of the quotients from (D 2).

D-3 h = g dr

The integrals for the single Tschebyscheff –polynomials are

107

from

The coefficients thus results in g (r ) function for an integral:

is determined corresponding to the boundary conditions.

D-4 h = dg/dr

For the derivation of the Tschebyscheff series, one obtains the reverse of (D. 3)

The starting values are and .

D-5

On one hand for (D 4), the calculation specification for

108

is specified from

And on the other side for (D .2) the following is obtained from

to

D-6 Differential operator of the second order

The operator for the semi implicit integration from (2.14) on one side and the r- and - derivative of the left side operators in (2.15) on the other side, can both be brought in the form

The relation for the second derivative gives a two-timed implementation from (D 4)

The combination of (D 4) and (D 2) provides for

In this context, insertion from (D 2) is then established for

One finally gets the preferred recursion relations through the combination of three precedents:

109

The is obtained from the linear system of the formula

The coefficients are placed in the first line of the matrix M that follows from the conversion of the boundary conditions at the wall. Coefficients from (D 6) are implemented from the third line. Because M (with an exception of the first two lines) is supposed to be penta-diagonal, it is valid for the line k = + 4 and for the following k = + 6, + 8,….. If an explicit boundary condition for the center of the pipe is fulfilled, then from it the outstanding coefficients are derived for the second matrix line. Alternatively one goes back to (D.6) with k = ( + 2) {2,3}, so this would contradict the restriction k ≥ 4. In this case, the total representation can thus be used instead of (D 5).

The resulting relation

can be used only with k = + 3.

110

E Formulation of boundary conditions in the radial direction

If a sequence should fulfill the following boundary value

then, these can thus be converted in the specification for the coefficients , in which one evaluates the Tschebyscheff polynomials i.e. derivatives at the corresponding radial positions. Here the points r = 1 (pipe wall) and r = 0 (center of the pipe) are noteworthy.

We get a Dirichlet-boundary condition at the wall due to

It is valid for a Neumann-boundary condition at the wall due to

An odd function generally disappears at the center of the pipe. An even function ( = 0) takes the following value due to

While an even function always possesses horizontal tangents on the tubular axis, the derivative of an odd function ( = 1) is given as follows due to

111

F Differentiation on the boundary

Compact differences represent approximate relations between the values of a function and its derivative at single grid point. An example for such type of a relation is a formula

One would like to implement it based on the differentiation of the function that lies on equidistant points j = 0,1, and thus one can insert it from j = 2. Two boundary points must be found on the other remaining linear independent equations, which can be read, in the supplement to (F.1) as an example.

Both these formulae are of the 8th order as in (F 1).

The actual attainable accuracy should be illustrated based on an example. It should be f(z) = sin(z) with the exact derivative f’(z) = cos (z). The grid size sums up to z = (2/, which corresponds to a release of 12 points every period. Both one-sided differences are used on the left boundary, the exact values are specified on the right boundary z = 2/ Figure 6.1 shows deviative of an exact solution in logarithmic applications. One knows in detail, that the errors rise to several bigger orders against the (left) boundary z = 0.

112

Figure F.1: Values of errors while calculating the first derivative.

113

G Finite differences at the inflow and outflow boundaries

The finite differences on the interval boundaries must necessarily differentiate themselves from the differences in the inner intervals, because the expansion jmax is limited to one direction. Differences of the lower order can be inserted in the boundary through the extension of area described for the differentiation in section 2.6.3.5 (P 19). This is concrete for the first derivative of the left boundary ( Δj ≥ 0 )

At the neighboring point ( Δj ≥ -1 ) the following is implemented

and for ( Δj ≥ -2 ) the formula

From here onwards the implementation from (2.24) will not stay in the way, till the reflections of the above-mentioned three differences are inserted at the right boundary. Analogously for the second derivative

The formula (2.27) can be used in the integration from the first unknown value through the already mentioned extension of the calculus area. The sizes are independent of z in the outlet current boundary, so that even a large disturbance error can be irrelevant and

can be carried out for J-2 j J with J as the highest index.

114

H Discretization of mixed operators

The operator

of the Poisson’s equation (2.15) can be separated into the components.

In the following, discretization of such mixed operators is described for a general case, whereby an approximation with Tsechebyscheff-series is assumed in radial (r) and with compact finite differences in the axial direction (z).A function is approached by an area through . A linear operation

as well as section D (P 105) is executed and specified in the series as

Thus, the coefficients can be determined at the same time from the relation of the formula

A linear operation is generally written as

Based on the linearity and thus

A determination equation can be obtained for the new coefficients , by taking the weighted sum of (H 1) and (H 2) with the other coefficients

115

And both these expressions are added:

corresponds to the given right side of the differential equation, is the desired solution.

While the discretization in the radial direction with Tschebyscheff-polynomials is already spectrally exact, it is then valid for the compact differences such that a small discretising error is obtained with the consideration of more grid points in the neighboring point j. If one maintains a fixed value |Δj|max at the time of displaying an algorithmic solution, then one limits αz and βz in a similar manner to the right hand side of the above equation and this is done by mixing the coefficients. If |Δj| ≤2 is given as an example then the lowest attainable discretization error is of the eighth order for the second derivative, which is relevant here. This is also achieved with (2.26).

116

I Diagonalization of a Toeplitz-matrix

The algorithm illustrated in the following is to an extent the generalization of what is described by an example from Swarztrauber (24). This generalization is valid to an extent that the system is not executed only with tri-diagonal Sub matrix. A simple representation of part of the single dimensional system

Tv = b (I.1)

only with a single penta-diagonal Toeplitz matrix

of the size ( N + 1) x ( N + 1) and has the vectors

of the length N + 1.This system can be to expanded to

117

with the circular matrix

and

The inner rectangle in C marks the unchanged matrix T. The dimension of C depends on the discretization and in the existing case it is a pentadiagonal matrix (N + 5) x (N + 5)

If one writes the eigenvectors of C as matrix, then this matrix is identical with the operator of the discrete Fourier-transformation (5)

c ≠ 0 is an arbitrary constant. Both operators diagonalize C:

λk are the eigenvalues of C. The solution of (I. 2) is

with the inverses .The first step for the calculation of the unknown v is thus an inverse transformation of the

right hand side that can be carried out efficiently by means of “Fast Fourier transformation” FFT (25). Whereas in the single dimension case, the weightage solely takes place with 1/λk , in two-dimensional algorithm, the decoupled system of discretization in the radial direction is solved with the help of LU-Analysis (refer to e.g. 18). Finally a further Fourier-transformation is carried out.118

The coefficients bj, j S = {-2,-1,N + 1, N + 2} are initially unknown. On the other hand the values vj, j S must correspond to the specified boundary conditions. The system (I.2) is first run through flow with arbitrary values . The product is calculated from a one time calculated influence matrix and the error ∆v j at the boundaries then gives the correct values . The solution obtained in the second run is thus similar to the solution for (I.1)

119

J Calculation of the Influence matrix

For the concrete procedure, the relation between the pressure at the wall on one hand to the derivative of the speed normal to wall on the other hand is to be likewise identified. As through the discretization, all the amounts are given only as discrete levels zi + jz, it deals with a linear connection between two vectors, which can be illustrated as a matrix. The initialisation of such Influence matrix goes exactly through the steps of the integration procedure.

One begins with the specification of a pressure distribution at the wall. In order to separately determine the influences the individual , this distribution is a vector, whose element j has the value 1, while all other elements are set to zero. With this condition the homogeneous Poisson equation is solved.

This solution incorporates momentum conservation equations. Since the speed is only expected to effect the pressure change, the other explicitly integrated terms are omitted.

The appropriate Runga Kutta indexing steps (see section B (S.100) ) are .

Further advantage of the fact that all the coefficients a ii of the integration scheme have the same value, is seen here. Thus the result is proportional to any selectable coefficient and can be used for all indexing steps after being appropriately scaled.

If, is determined as (2.6), then the derivative can be calculated. These steps are carried out for all the points of the wall . The ascertained derivatives are shown as columns of a matrix Mn, which thereby fulfils the equation

120

K Stability solutions

K-1 Equation system

In the scope of the linear stability theory, small disturbances (infinite times) of temporarily constant base flow are investigated. As per this pre-requirement, the non-linear terms of the Navier-Stokes-equation can be ignored from smaller to higher order. The unknown functions

are formed as linear combinations of complex sizes

The partial differential equations are transferred in a usual way with r as the only independent variable. The exponents are derived in the section C (P 102). The solutions must be periodic in the azimuthal direction, hence in . Thus only for n, real, integral numbers come into question. The axial wave number and the frequency

on the other hand can accept complex values. Based on its physical meaning in section,

(Incomplete sentence in German).

A linear system is thus solved

where operator possesses only real coefficients. Thus is also a solution of this system:

Based on this fundamental one can comply through the superposition of the demand that the velocities1, and the pressure must be purely real. The real portion again illustrates a solution namely

1 Only and are meant here, because are complex in physical space.2 This analog is valid for the imaginary part .

121

The continuity equation (2.2) with the statement (K 1) is

or

If one linearizes the momentum conservation equations (2.5) around the base flow (2.4) and introduces (K 1), one gets,

with

122

Analogously the Poisson equation becomes

In the general case , the sizes are calculated directly from (K 4d) and (K 5). The components and reconstruct themselves according to

its counterpart (2.6). results from (K 3).

K-2 Boundary conditions

The no slip conditions at the wall for the three velocity components is

From the first two equations, both the conditions are obtained

From all the three and the continuity (K 3), the following results are obtained

The condition (K 7) could be transferred alternatively, wherein one gives the polynomial for u in the statement (K 1). The equations become even more complex, besides

(K 8) is predetermined explicitly.

K-3 Symmetry

In (K.1) and correspondingly in (K.2)-(K.5), n is replaced by -n, hence one can establish the original differential equations, wherein one sets :

123

Because all the boundary conditions (K6 –K8) are homogeneous, both the solutions can differ from each other only through constant factors, thus

One can thus form further linear combinations. If one sets , thus3

Corresponding to case , is set to .

K-4 Special case

In case n = 0, the differential equations (K 4d) for and are identical. Because the boundary conditions (K 7) are homogeneous, both the functions must be linearly dependent. It will be clear, as to which of the dependencies are possible, if one of the returns to the original variables and from u. The linearized equations (K 4b), (K 4a) and (K 5) are then

3 The physical values of and p are considered even here.

124

While the second and the third equations are coupled together, the first is an independent determinant equation for , because and are represented respectively, . One gets solutions for with on one hand and solutions and with on the other.

K-5 Special case

The Poisson equation reduces itself for = 0 to

The general solution of this equation is . As must be finite, . Hence can be introduced into the derivation of the further

equations. and are directly coupled over the continuity equation in the (K 3) formula,

because the term falls out. If one returns to the primitive variables, then (K 2) in the present case is

and the momentum conversation equations

with the boundary conditions

is , and can nevertheless be . The reverse is not valid, because (K 12) can be forced in case of and . Thus one gets solutions either with disappearing or with non-disappearing .

If one multiplies the differential equations with Re i.e. Re2, one gets the system

125with

It is evident in this notation, that a known solution can be scaled for a Reynolds-number Re = Re1 for Re = Re2 ≠ Re1, in which one sets

with ρ = Re1 / Re2

K-6 Special case n = α = 0

If both n as well as α are set to zero, then the equation for pressure is

Its solution is general in r. Thus only . is again valid. A spatially constant pressure arises physically that varies temporally according to exp(-iβt). Such a perturbation pressure does not have any influence on the velocity and one can set it to .

The continuity (K.2) becomes

The solution is only for C = 0 and is thus analytical. The remaining, de-coupled equations are

with the boundary conditions

K-7 Temporal stability

If one selects the definition areas of the parameters in (K 1) as

126

the disturbances in axial and azimuthally direction are periodic. According to the sign βi they temporally disappear from (βi < 0) or grow exponentially at (βi > 0). The system is therefore stable with regards to disturbances with negative βi. If there is at least one disturbance with positive βi, then the system is unstable. The solution methods applied here for the system with de-coupled equations lead to a linear eigenvalue problem of the formula

with the frequency β as eigenvalue, quadratic matrices A and B and a vector w of the coefficients of the discretized unknown functions. Elements from A and B result from the discretization that are described below as functions , n, and Re.

K-8 Spatial stability

If one alternatively uses

,then the functions are temporally periodic and the imaginary portion αi of the wave number determines, whether they disappear (αi > 0 ) or grow (αi < 0 ). While the equations (K 4) and (K 5) are linear in β, the coefficients (i)2 are non-linear in , through the second derivations as per z. Also for the investigation for the spatial stability, a linear eigenvalue problem of the formula should again be set up

According to Bridges and Morris (2) as well as Tumin (26), and are additionally introduced to the existing and , so that the second derivative in the main flow direction can be represented as the first derivative of this new unknown

127

with . The momentum conversation is thus

and the determinant equation for the pressure is

K-9 Discretization

By way of approximation, the functions are represented by Tschebyscheff-series

At the same time Index s stays as a representative for one of the unknown, hence for example ., etc.

K-9.1 Differential equations

The established linear, standard differential equations can be put forth in the general form

with the respective eigenvalue . The constant coefficients and are functions of n and Re as well as for the temporal or for the spatial stability. Coefficients of the differential statements can be represented depending on the coefficients as series

Individually the following is required:

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These formulas can be found in an easily changed form with Gottlieb and Orszag (7).Tschebyscheff -polynomials form an orthogonal system. If one uses (K 20) in (K 19),

one can carry out comparison of coefficients, and get

K-9.2 Boundary values

The general form of a boundary condition is

whereby in the existing case, the right hand side always disappears. Derivatives at a radial position are obtained from a weighted sum of coefficients

Especially in the existing case, one has

Generally, the discretized representation of a boundary condition is thus as follows:

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K-10 Structure of matrices

K-10.1 Temporal stability

K-10.1.1 General case

For the equation of temporal model, both the matrices A and B from (K.17) are sub-divided each into 3x3 square sub-matrices of dimension (L + 1) x (L + 1). The sequence of the unknown coefficients in vector w is basically arbitrary. Here the following is selected:

The first L lines of the sub matrices are obtained from the differential equations. The respective last lines are replaced by discrete equivalents of the boundary conditions

At the same time I stands for identity and

for the matrix , whose lowest lines are exchanged against . The single elements are

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are the operator matrices from (K 20), are the line vectors of the coefficients from (K 21).

K-10.1.2 Special case

One puts forth

one thus gets the same matrix B for the equations from the section K-4 (P 123), the matrix on the left side and its elements are however

131

The system disintegrates into two independent partial systems for v on one hand and vr and p on the other side.

K-10.1.3 Special case = 0 n 0

Likewise with the vector

the equations are again obtained from the section K-5 (S 124) as well as the above-mentioned matrix B

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K-10.1.4 Special case n = = 0

The equations from the section K-6 (P 125) are finally obtained with the vector

and both the matrices

K-10.2 Spatial stability

In the case of six unknown functions, each of the matrices A and B from (K 18) consist of 6 x 6 sub-matrix. Vector w of the unknown coefficients is

Sub matrices consisting of the first, second and the fifth lines are pre-determined through the equations

133

The remaining group of lines result from differential equations and the boundary conditions:

134

The corresponding vector for n = 0 is

Again B remains unchanged for n 0, A on the other hand is as follows

135

K-11 Numeric solution

The eigenvalue problems (K 17) or (K 18) can be solved with a routine from a program library like LaPACK. One can repetitively exactly calculate single pairs of eigenvalues and the associated eigenvectors. The specified algorithms in (9) consist of three steps: With an approximate solution (k is the index of iteration steps) one calculates

from an improved eigenvector . Then a new approximation for the corresponding eigenvalue is

where is the conjugated transposed vector of w. where is the conjugated transposed vector of w. Because the matrix is nearly singular, the standard of is very high. The iteration instruction is thus determined by standardization, approximately

In the existing case all elements of B disappear outside the main-diagonal. The above-mentioned iteration thus becomes

K-12 Results

Figures K 1 and K 2 compare the results of a specific parameter replacement at different lengths of Tschebyscheff –series.

136

Figure K 1: Numeric approximation of the most unstable eigenvalue β for the temporal model in the complex level for n = 1, Re = 2280 and α = 1 L = 30, L = 50

.

137

Figure K 2: Numeric approximation of the most unstable eigenvalue β for the spatial model in the complex level for n = 1, Re = 2280 and β = 0.96 L = 30, L = 50.

138

Figure K 3: Sums of the eigenfunctions and for the spatial theory for n = 1, Re = 2280, = 0.96 and 1.06360 + 0.05123 i (above) and/or 1.15042 + 0.34805 i (below).

The most unstable eigenvalues and are respectively identical for both the values of L. Eigevalues with bigger sums of the imaginary parts correspond to strongly damped and more complex eigenfunctions, which in the case of rough discretization are not reproduced that well. The results therefore diverge for L = 30 and L = 50 in decreasing i or increasing i.

Figure K 3 shows exemplary sums of eigenfunctions for the unstable and strongly damped eigenvalue.

139

K-13 Secondary stability analysis

K-13.1 Pre-requisites

Not only a laminar Hagen-Poiseuille profile (2.4 a), but also a deformed base flow can be examined for stability. Within the scope of this work it is assumed in a simplified manner, that the base flow

does not contain any radial or azimuthal components and does not vary with the axial coordinate z. The deformation is retained by physical strength

K-13.2 Differential equations

The statement (K 1) becomes the series

The analytical equation is expanded by additional convective terms:

The linearized terms (2.9) of the Poisson equation (2.8) are

140

After consideration of the continuity (2.2), the equation for pressure is then

and with the statement (K 22)

K 13.3 Symmetry

One replaces n by -n in the figures (K 23) and (K 24) and exchanges with , thus one again gets the original differential equations because of , whereby is exchanged with as well as with :

141

As the boundary conditions (K 7) and (K 8) are homogeneous, the solutions can differ from each other only by constant factors, thus

The goal is to determine the constants C + and C - so that one can substitute some of the unknowns through others and can therefore reduce the dimension of the calculated eigenvalue problems. As far as C 0, the following quotient is desired.

If one considers (K 26) for n = 0, one obtains velocities

This system has a non-trivial solution, if its determinants disappear, thus

In the case of n 0, we gets the following for n from (K 26)

thus

in the case of final pressures.

Since the equations are de-coupled over the folding sums of the convective terms, relations between different ncan be developed. One weighs (K 23) with and (K 25) with

(above signs respectively) and subtracts the resulting equations from one another and hence the following remains

If the coefficients of the base flow do not disappear, then in the case of a non-trivial solution each of the item must become zero:

Together with the definition of the coefficients (K 26), one gets the system

142

If all the determinants should disappear, then all the quotients Q n must be same. Together with (K.27), only two variants remain

The first case “’ +”’ implies

Thus,

The second “’-“’ implies

Thus

Only and are selected as the remaining unknowns, whereby the coefficients with indices n ≥ 0 or n > 0 are sufficient for the pressure.

K-13.4 Special Case = 0

In the case of = 0, a separate observation is required as in K-5 and K-6. The Poisson equation (K.9) and (K.14), the continuity (K.10) and (K-15) as well as the momentum equations (K.11) and (K.16) can be adopted unchanged for the corresponding indices ‘n’. If one removes the physical strength

from the momentum balance for the perturbation numbers in axial direction, then the following remains

143

As and over (K.10) or (K.15) are coupled with each other, the items

of the convective terms can be simplified. For n = 0, disappears, thus

According to (K.10), for n ≠ 0

And thus for n’ ≠ 0

If one considers the numbers as the unknown searched numbers, then the equations will finally have real coefficients. Thus, the phases of these unknowns are equal and independent of r. The eigenvalues are purely imaginary. Thus one can limit itself to purely real

as well as purely imaginary values . However a phase relation of eigenvectors qualitatively changes the eigenfunctions, as finally from the complex eigenfunctions (K-1), only the real portions are taken into consideration. This is also the case with 0, where all the possible constellations appear through continuous phase change exp (iz) during a period 2/. In order to illustrate all the aspects of the eigenfunctions for ≠ 0, even the imaginary portions are evaluated, apart from the real ones of the eigenfunctions. Each superimposition of these two functions is however, again a solution.

In the above equation, one can again replace n by – n and obtain from the comparison

However the coefficients of the indices –n and +n are not coupled. As the base functions of the eigenfunctions (real portions) or (imaginary portions) are identical, one can limit himself to n 0.

The scaling (K.13) for a variation of the real Reynolds numbers is also valid here.

144

K-13.5 Calculation of the base flow terms:

If is available in the form of Tschebyscheff series, one can calculate the factors or with the help of recursive application of (D.1). The product from these and the

perturbation numbers result out of the general product of two even Tschebyscheff series

As in (K.29), if the product of a derivative 1/r /dü/ dr is multiplied with the base flow, then as in section K.9.1 (P-127), one replaces the product through

145

L Determination of kinetic energy and the energy flow

L-1 Fourier analysis in axial direction: The calculated data is available in the spectrally analyzed form for the azimuthal and radial directions. If one can periodically continue the perturbation numbers of a specific section between two cross-sectional planes z = z0 and z = z0+ then one can also carry out a Fourier analysis in z. The procedure can be described with the help of

which is periodically present in z, with the wavelength . If one chooses the interval z0 z <z0

+, for the analysis, then

with = 2/, =z’ and the coefficients

If the data is available on discrete, equidistant points

Then

If u is real, the coefficients must fulfill specific conditions. If one groups with

and , the items of analysis are

146

147

So one knows that

must be valid and especially the is real. Over and above, u can be represented as pure cosine series in , thus

The is then purely real, so

Similarly for the case of sine series

is valid with pure imaginary and the disappearing

148

L-2: Definition of kinetic energy

The following formulation summarizes the symbols used.

The spatially shown kinetic energy of the flow is

with . The square of the velocity vectors combines the squares of its three scalar components according to

However the energy can be split into components

149

L-3 Integration over and z

The order of the integration steps in (L-30) is arbitrary. If one starts with the integration over , for the scalar one obtains

The following estimation for the axial analysis results into

Thus equation

L-4 Integration over r

There remain only integrals of the form

To be calculated with

150

On the basis of quadrature of a non-linear operation, it is more efficient to make an estimation in the physical space. For that the function u(r) is illustrated on the collocation points

The function value on these points is

Now directly the half sum square can be formed. The function U = ½ u2

is real and even. Therefore its spectral illustration is

With coefficients

Thus

With the integral

results the following weighting factor

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L-5 Treatment of the pressure

With corresponding initial and boundary conditions in the axial direction, the velocity can be periodic and this fulfils the initially placed condition. This is not valid for the pressure. The pressure of the base flow (2.4b) is a function, which falls highly monotonously in z. In the momentum conservation equation (2.3) only the inducted force field enters, which can be completely periodic, when valid.

p < 0 is thereby the additional pressure loss on the basis of the perturbation flow. The periodic portion of the disturbance pressure is then

and the pressure force is

L-6 Temporal development

An equation is searched which describes the change in the kinetic energy in the periodic process. Thus

The time derivative of the velocity square can be traced back to the derivative of the velocity.(For clarity the indices are ignored half way)

Thus

152

With the help of projection, one gets the equation for components from the momentum conservation Equation. The following coefficients evenly results for a linear item w

For the product of two terms u and v, one obtains

Then the three scalar equations are

153

with

154

M Karhunen-Love analysis

The scalar product of two velocity fields and is defined as

If the volume considered is tube portion between two axial positions z0 and z0 + , then

If the velocity in azimuthally direction is given as a Fourier series

Then

Further, if both the functions are given on discrete points zj = z0 + jz then

The integration of r as in section over (sentence incomplete in the German text)

The procedure to calculate the Karhun-Love eigenfunctions as described by Sirovich [23], places a matrix from a given equation of periodic equidistant velocity distribution

The eigenvectors ci of these matrices also the solution of the eigenvalue problems

illustrates the weighting factors for the determination of the eigenfunctions Equation

155

As the matrix is symmetric, its eigenvalues i are real and the eigenvectors are orthogonal [3]. From this follows the orthogonality of the eigenfunctions

The reconstruction of the velocities through the superimposition of the eigenfunctions

whereby the coefficients can be calculated from the projection

The spatially shown kinetic energy at the time point tm is

Analogously the energy of the eigenfunction is

On the basis of orthogonality results the total energy as weighted addition of 156

individual energies.

Each eigenvector ci can be standardized with an arbitrary constant. This is determined in such a way that the contribution of the eigenfunction to the temporally shown energy

is equal to the energy of the eigenfunctions, hence

So the searched c i allows the arbitrarily standardized eigenvectors cI * to win over

The sign can then be so selected that the coefficients in the center are positive. From this standardization, the following is obtained:

The orthogonality of coefficients i can be easily shown over

157

N Galerkin-projection of the vortex equation

The vortex equation (7.4) in the scalar form is

with non-linear terms

The components of the eddy strengths are

From those eddy strengths can be formed which corresponds to Karhun-Love eigenfunctions (7.2) and one can insert this Equation

in (7.4)

For the Galerkin-projection test functions, is used as a test function, which leads to the usual differential equation.

158

If one defines

One obtains the compact style

If one forms an inverse T-1 for the square matrix T with

then one obtains individual equations for the periodic changes of the coefficients

Change of kinetic energy (M.1) is thus

If one shows this over time and considers the orthogonality (M.2), then one obtains

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Resume

Personal Data

Name: Jörg Karl Christian ReuterBirth date: 15th May 1968Birthplace: WürzburgNationality: GermanMarital status: SingleEducation9/74 - 10/77 Elementary school Unterföhring10/77-7/78 Elementary Dettelbach

9/78 -10/82 Egbert-Secondary school, Münsterschwarzach10/82 -6/87 Markgraf-Georg-Friedrich-Secondary school, Kulmbach1/85- 3/85 Devonport High School, Plymouth

Civil service

7/87 -10/87 Youth hostel, Tannenlohe11/87 -2/89 City- and district hospital, Kulmbach

Study

10/89-7/95 Mechanical engineering, University of Karlsruhe; Speciality: Fluid mechanics, Measuring and control engineering10/92 -6/93 Imperial College, London; Study: "`Adaptive methods for transient flows"'9/94 -6/95 Ecole Nationale Suprieure d'Arts et Mtiers, Paris und Lille; Thesis: "`Identification et commande d'un systme multivariable"'

Professional activity

8/95 Scientific co worker at the Institute for Aerodynamic and Gas dynamic, University Stuttgart8/98 12/98 Research fellow at the Cornell University, Ithaca Stuttgart, October 10, 2002