Appendix A: The Fully Developed Velocity Profile for Turbulent Duct Flows

Transcription

1 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows This appendix discsses the hydrodynamically flly developed velocity profile for pipe and channel flows. The geometry nder consideration is shown in ig. A. ig. A.: Geometry and coordinate system or the flow in a parallel plate channel, the velocity components in the x, y and z direction are, v and w, whereas for the flow in a circlar pipe, the velocity components, v and w denote the flow in the x, r and ϕ direction. We restrict or considerations to an incompressible flow with constant flid properties. The Reynolds averaged Navier-Stokes eqations and the continity eqation are given for a flow in a parallel plate channel by (Kays and Crawford (993 Parallel Plate Channel ρ p p v μ ρ ρ x yy x x y x y ρ p p v v v v v μ ρ ρ y x y x y x y v = x y (A. (A. (A.3

2 4 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows or the flow in a circlar pipe with rotational symmetry, the eqations are given by (Kays and Crawford (993 Pipe p μ ρ v r xx r x r r r μ x ρ ρ r r x vv v p μ v v ρ v r μ x x r r r r r x ρ ρ ww ρ r r x r (A.4 (A.5 = x r r (A.6 or a hydrodynamically flly developed flow, the axial velocity component does not change (by definition with the axial position and therefore / / x is zero. If we introdce this reslt into the continity eqation (A.3 or (A.6, we obtain that the radial velocity component v is eqal to zero and the continity eqation is then atomatically satisfied. The fact that v = for the flly developed flow is obvios, as a non-zero velocity component v wold lead atomatically to a change in the axial velocity component for different axial positions. Introdcing these reslts into the above given eqations reslts in Parallel Plate Channel p d = μ ρ x dy d( p d ( = ρ y dy dy (A.7 (A.8 Pipe p d = d ( x r dr r d dr r dr (A.9

3 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows 5 p ρ d w w = ρ r r dr r (A. The above eqations have to be solved together with the following bondary conditions: Parallel Plate Channel d y = :, dy y h : (A. Pipe d r = :, dr r R: : (A. In the above eqations, the partial derivatives for the axial velocity component and the trblent stresses have been replaced by total derivatives, becase the fnctions are only dependent on the coordinate orthogonal to the flow direction. The eqations describing the hydrodynamically flly developed flow in the parallel plate channel and in the pipe are qite similar, as it can be seen by comparing Eq. (A.7 with (A.9 and Eq. (A.8 with (A.. They can be written in a condensed form by sing a flow sperscript. or the flow in a parallel plate channel =, whereas for the flow in a pipe =. This reslts in the following eqations: p x d r dn d dn ' v ' (A.3 d p d( ( = ρ dn r n with the bondary conditions d n = :, dn n L: : (A.4 (A.5 In these eqations, n denotes the coordinate orthogonal to the flow direction (n = r for the flow in a pipe and n = y for the flow in a parallel plate channel. The qan-

4 6 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows tity L = R (pipe radis for pipe flow and L = h (half channel height for the flow in a parallel plate channel. The Eqs. (A.3-A.4 are solved together with the bondary conditions given by Eq. (A.5. Eq. (A.4 can directly be integrated and reslts in p = p W R d r r dr (A.6 Eqation (A.6 shows nicely that the pressre in the direction orthogonal to the main flow changes across the dct de to the presence of the trblent stresses. or a parallel plate channel, the change in pressre is only cased by the normal trblent stress, whereas for a pipe flow also the difference between the normal trblent stresses in radial and circmferential direction cases an additional change. Differentiating Eq. (A.6 with respect to x leads to p W p = x x becase the trblent stresses only depend on the coordinate orthogonal to the flow direction. or the flows considered here throgh a pipe or a parallel plate channel, the mass flow rate is constant and the integral form of the mass continity eqation leads to L ( L r (A.7 dn where denotes the mean axial velocity in the dct. The above given eqation determines the yet nknown pressre gradient in the axial direction ( / x is a constant for the hydrodynamically flly developed flow. In order to solve Eq. (A.3, the trblent shear stress ρ v has to be related to the mean flow field. This can be done sing a simple mixing length model. The se of the simple mixing length model is adeqate for this type of flows (Cebeci and Bradshaw (984. However, for more complicated flows, like the flow throgh a pipe with cyclic constrictions, mixing length models may lead to inaccrate answers. The reader is referred to Reynolds (974, Lander (988 or Kays or Crawford (993 for a review of more advanced trblence models. Using the mixing length model, the trblent shear stress can be written as d v' ' ' =ε (A.8 m dn with the eddy viscosity ε m given by

5 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows 7 d dn ε m = l (A.9 The mixing length l has been measred by Nikradse (93 for pipe flows. He obtained the interesting reslt that l does not depend on the Reynolds nmber for Re D > 5. The mixing length distribtion can be approximated by (see Schlichting (98 4 l n = n L.6 L L (A. Near the wall, the mixing length distribtion, according to Eq. (A., approaches the one given by Prandtl (see Schlichting (98 l,.4 (A. Close to the wall, the trblent flctations are damped ot and, below a certain dimensionless distance away from the wall, the trblent flctations are zero. In order to se the mixing length distribtion throghot the whole calclation region, it is convenient to modify the expression sing a damping fnction This garantees that the mixing length tends to zero within the laminar sblayer. Cebeci and Bradshaw (984 modified the mixing length distribtion, given by Eq. (A., with the van Driest damping term (van Driest (956. This leads to l L with the abbreviations 4 = n n y 6.6 exp L L 6 y y ν W τ =, τ W τ =, W ρ (A. (A.3 Introdcing Eq. (A.8 into Eq. (A. reslts in an ordinary differential eqation for the axial velocity component p d = W r d d d μ ρl xx dn dn dn dn W (A.4 This eqation has to be solved together with the bondary conditions given by Eq. (A.4. The axial pressre gradient / x can be replaced by the shear stress ve- W Zagstin an Zagstin (969 developed theoretically a formla for the mixing length distribtion in a pipe by solving the conservation eqation for the trblent energy. Their mixing length distribtion is also in good agreement with the measrements of Nikradse (93.

6 8 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows locity τ. In order to do this, let s consider the forces acting on a small dct element (shown in ig. A. ig. A.: orce balance for a small dct element Eqating the forces acting on the small dct element shown in ig. A. reslts in: ( R W R x W, Pipe (A.5 h W x, Parallel Plates (A.6 rom the above two eqations, one obtains the following relation between the pressre gradient and the wall shear stress τ p τ W τ = = x L L (A.7 Introdcing this reslt and the following dimensionless qantities into Eq. (A.4 r r =, L n n =, L =, τ l l =, Re L τ = L τ ν (A.8 reslts in the following ordinary differential eqation for the axial velocity component d τ d d Re r l Re τ dn dn dn (A.9 Integration of this eqation and application of the bondary condition d dn = for reslts in l Re d τ d Reτ n dn dn (A.3

7 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows 9 After solving for the nknown velocity gradient one obtains from Eq. (A.3 d = dn Re 4 Reτ τ (A.3 This ordinary differential eqation has to be solved nmerically with the bondary condition that = for. In addition, the continity eqation in integral form, Eq. (A.7, has to be satisfied. This eqation reads after introdcing the dimensionless qantities according to Eq. (A.8 Re Re L τ = r dn, Re = L L ν (A.3 or a selected Reynolds nmber Re L, a vale for Re τ has to be estimated. After this, Eq. (A.3 is solved nmerically and the reslting velocity distribtion is inserted into Eq. (A.3. rom Eq. (A.3 a new vale for Re τ is obtained and the iteration is carried on p to the point, where both Eqs. (A.3-A.3 are satisfied. ig. A.3 shows a comparison between predicted and measred velocity profiles in a planar channel and in a pipe. ig. A.3: lly developed velocity profiles in a planar channel and in a pipe It can be seen that the application of the simple mixing length model leads to qite satisfactory reslts for the flly developed flow in a pipe and in a parallel plate channel.

8 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows Note that also the hydrodynamic developing flow in a dct can be predicted as well by sing a modified mixing length model. The reader is referred for more details to Cebeci and Chang (978 and Cebeci and Bradshaw (984. Antonia and Kim (99 compared calclations based on a mixing length model for hydrodyamically flly developed flow in a channel with experiments and also with direct nmerical simlations (DNS. They fond good agreement between their calclations and the DNS data. After establishing a simple calclation method for the hydrodynamically flly developed flow, the niversality of the velocity profiles for trblent pipe flows is addressed next. As it can be seen from Eq. (A., the total shear stress τ rx for the pipe flow consists of a moleclar part and of the trblent shear stress: d τ rx (A.33 dr After integrating Eq. (A. it can be seen that the total shear stress τ rx varies linearly across the pipe. With this reslt, one obtains from Eq. (A.33 d τ - = / rx W W / R = (A.34 dy ε ε ν ν where y r W R. Eq. (A.34 has been the basis for many considerations for trblent pipe flows, which led finally to approximations for the flly developed trblent velocity distribtion. In the following section, we describe mainly the work of Reichhardt (95 on this topic. The Law of the Wall Near the wall, the eddy viscosity distribtion ε m can be approximated relatively easy from experimental data. Nikradse (93 derived from his experimental reslts the following approximation for ε m εm = κ y ν y y ν W τ, = (A.35 Introdcing Eq. (A.35 into Eq. (A.34 and integrating the reslting expression for the region close to the wall ( / W and y reslts in = ln c κ y (A.36 The logarithmic velocity distribtion in Eq. (A.36 has been derived nder varios assmptions by Prandtl (95 and von Karman (939. It is a very important reslt for trblent flows, becase of the niversality of this velocity distribtion for varios trblent flow fields. rom Eq. (A.35, it can be seen that this eqation looses its meaning, if we consider vales of y very close to the wall, as for

9 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows y < the trblent exchange in the viscos sblayer of the flow tends to zero. In order to overcome this difficlty, Reichardt (95 developed the following distribtion for the eddy viscosity near the wall ε m ν y y y arctan y, (A.37 This fnction goes to zero for. or larger vales of y, Eq. (A.37 approaches continosly Eq. (A.35. Introdcing Eq. (A.37 into Eq. (A.34 and integrating, reslts after restricting again to the region close to the wall ( / W and y in y y y.5ln( 5ln( 7.8 ep exp ep exp y y 3 (A.38 Eq. (A.38 compares favorably with experimental data in the near wall region (see Reichardt (95. The Core Region In the central part of the pipe, all distribtions have to be symmetric to. Therefore, it is sefl to analyze this area with Eq. (A.34 after introdcing the radial coordinate instead of the wall coordinate. rthermore, the moleclar viscosity can be neglected compared to the trblent eddy viscosity in the denominator of Eq. (A.34. This leads to d dr = Re τ r ε ν (A.39 In the central region of the pipe, the eddy viscosity can be approximated according to Reichhardt (95 by εm κ = Reτ ν 3 r (A.4 Eq. (A.4 was fond to be in good agreement with experimental data. If we introdce Eq. (A.4 into Eq. (A.39 and integrate, the following reslt for the velocity distribtion is obtained C τ r = ln κ r (A.4

10 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows where C denotes the maximm velocity in the pipe at. Eq. (A.4 is valid throghot the flow area, except the region close to the wall, where the moleclar viscosity plays an important role (Reichhardt (95. An Approximate Velocity Distribtion for the Whole low ield Based on the two approximate velocity distribtions, Eq. (A.38 and Eq. (A.4, Reichhardt (95 developed an approximate velocity distribtion, which is valid throghot the flow area. 3 ( = ln y y y exp ep ep exp κ r 3 y y (A.4 with the constant C = / lnκ. Eq. (A.4 agrees very well with experimental reslts of Nikradse (93 and Reichhardt (94. Dring the past 9 years, a large nmber of different expressions for the niversal velocity distribtion in a pipe have been developed by different researchers. All eqations are based on slightly different assmptions for the eddy viscosity distribtion in the pipe and all the eqations describe more or less accrately the flly developed velocity distribtion. Table A. gives an overview over several different expressions for the velocity distribtion in the pipe. Strictly speaking, the above derived niversal velocity distribtion is only 5 valid for large Reynolds nmbers ( ReD. However, it was fond that they approximate very well the velocity distribtion also for mch lower Reynolds nmbers. Rothfs et al. (958 and Dwyer (965 showed that the inflence of the Reynolds nmber on the niversal velocity distribtion f can be incorporated by sing the following qantities y y /, / C C Pipe Pipe (A.43 In addition, Rothfs et al. (958 and Dwyer (965 were able to show that the modified velocity distribtion f can also be applied to the flow in a parallel channel and in a concentric circlar annls. The agreement between experimental data and the velocity distribtion f given by Rothfs et al. (958 for a wide range of Reynolds nmbers is good.

11 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows 3 Table A.: Universal velocity profiles for hydrodynamically flly developed dct flows Reference nctional dependence f f( ( Prandtl (9 y Validity range y.5 Taylor (96.5ln 5lny 5.5 y >.5 von Karman (939 y y 5ln y 3.5 y 5 5 y 3.5ln 5lny 5.5 y > 3 Reichhardt (95.5ln 5ln 78( 7.8 exp( / ( / exp( / 3 for all y y Deissler dŷ (954,.4 [ exp( ˆ] y ln y 3.8 y > 6 Rannie 4.53tanh (956.5ln 5lny 5.5 y 7.5 y > 7.5 Spalding (96 y =.8[exp(.4.4 (.4 (.4 3 (.4 4 ]! 3! 4! for all y Notter and Schleicher ( tan ( ln.5ln y C = c f 8 f W y y < y y W =.5 W.5 y W

12 4 Appendix A: The lly Developed Velocity Profile for Trblent Dct lows Kays and Crawford (993 y 5ln y 3.5 y 5 5 y 3 Reichhard (95 5ln.5ln.5( y n n y 3 Mirshina and Ogisio (97 /3 ln 3 6 /3 A R / /3 /3 y y / A π A tan /3 /3 3 Re τ Re τ y A 3A /3 6 y y.5ln 5lny 5.5 y y y < y y y y y y 5.5,.7Re 7R.5ln τ Re τ y =.3Re τ y < y Re τ y 4.4, 6.3 Re τ A 3

13 Appendix B: The lly Developed Velocity Profile in an Axially Rotating Pipe In the following appendix, the hydrodynamically flly developed velocity profile for a trblent flow in an axially rotating pipe is discssed. The geometry nder consideration is shown in ig. B. ig. B.: Geometry and coordinate system (Reich and Beer (989 or the flow in the axially rotating pipe, the velocity components, v and w denote the flow in the x, r and ϕ direction. or the following analysis, we restrict or considerations to an incompressible flow with constant flid properties. The Reynolds averaged Navier-Stokes eqations and the continity eqation, assming rotational symmetry, are given for example by Rothe (994 ρ ( ( p μ r μ x r r x r r r x ρ ρ r r x (B.

14 6 Appendix B: The lly Developed Velocity Profile in an Axially Rotating Pipe w ρ r μ x r r r r r r r x ρ ρ p μ vv v v ww ρ r r x r ρ ρ vw ρ r r x r r vw μ w w w ρ r μ r r x r r r r x r (B. (B.3 x r r = (B.4 or a hydrodynamically flly developed flow, the axial velocity component does not change with the axial position and, therefore, / / x is zero. If we introdce this reslt into the continity eqation (B.4, we obtain for this case the reslt that the radial velocity component v is eqal to zero and the continity eqation is then atomatically satisfied. The fact that v = for the flly developed flow is obvios, becase a non-zero velocity component v wold lead atomatically to a change in the axial velocity component for different axial positions. Introdcing this reslts into the above given eqations reslt in ρ ρ ρ = r r r r ρ r w p ww (3.3 w r 3 μ r ρr r r μ r r r (3.4 with the bondary conditions p x r r μ (3.5 ρ r r :,, r r R: :,, W (3.6 As mentioned in Chap. 3, experimental data show that the tangential velocity distribtion is niversal and can be approximated by Eq. (3..

15 Appendix B: The flly developed Velocity Profile in an axially rotating Pipe 7 w r = w W R W (3. igre 3.4 shows the tangential velocity distribtion for different rotation rates N as well as for different Reynolds nmbers. rom ig. 3.4, it is obvios that Eq. (3. is a very good approximation for the tangential velocity distribtion. The axial velocity distribtion (r can be calclated from Eq. (3.5. In order to do this, the trblent shear stress in this eqation has to be related to the mean velocity gradients. This can be done by sing a mixing length model according to Koosinlin et al. (975. This reslts in the following expression for the trblent shear stress / = ρ w l r = ε ρ m r r r r r r ρv ε ρ (3.7 ig. 3.4: Tangential velocity distribtion (Reich and Beer (989 where the mixing length distribtion l is given by (Reich and Beer (989 l l = Ri 6 (3.8 The mixing length distribtion l is the one for a non-rotating pipe (see Appendix A, Eq. (A.. The Richardson nmber in Eq. (3.8 describes the effect of pipe rotation on the trblent motion and is defined as

16 8 Appendix B: The lly Developed Velocity Profile in an Axially Rotating Pipe w Ri = r r w r rr r r (3.9 Withot rotation Ri =, there exists a flly developed trblent pipe flow. If Ri >, i.e. for an axially rotating pipe with a radially growing tangential velocity, the centrifgal forces sppress the trblent flctations and the mixing length decreases. If we introdce the expression for the trblent shear stress Eq. (3.7 and Eq. (3. into Eq. (3.5, we obtain the following nonlinear ordinary differential eqation after integration / Re D d D d d N r dr Re τ dr dr Re τ where the following dimensionless qantities have been sed r l r,,, τ W /, R R R D Re τ, ReD, N = ν ν τ w W (B.5 (B.6 In Eq. (B.5, the partial differentials have been replaced by ordinary differentials, becase the axial velocity is only a fnction of the radial coordinate. Eqation (B.5 is a strongly nonlinear ordinary differential eqation and has to be solved with the bondary condition ( (B.7 Eqation (B.5 together with Eq. (B.7 can now be solved in the following way: for a given vale of the Reynolds nmber Re D, a vale of the shear stress Reynolds nmber Re τ is assmed. After this, Eq. (B.5 can be integrated nmerically so that the bondary condition according to Eq. (B.7 is flfilled. After the integration, the continity eqation in integral form ReD 4Re τ = rdr (B.8 has to be satisfied. If Eq. (B.8 is not satisfied, a new vale for Re τ can be calclated from the eqation. After some iteration, the desired velocity profile is obtained. The reslting axial velocity distribtion is shown in ig As it can be seen from ig. 3.5, the agreement between the experimental data and the nmerical predictions is good.

17 Appendix B: The flly developed Velocity Profile in an axially rotating Pipe 9 ig. 3.5: Axial velocity distribtion as a fnction of the rotation rate N (Reich and Beer (989 or trblent flow in a non-rotating pipe, it is well known that the axial velocity profile can be described by an niversal velocity profile, given by C = (B.9 C τ In this eqation, C denotes the axial velocity in the pipe center. The velocity law is valid over a large portion of the pipe radis and the fnction f is not dependent on the Reynolds nmber. If one examines Eq. (B.5 in more detail, one sees that only one qantity describes the inflence of the rotation on the velocity field. This qantity is Re c D f Z N N/ Re 8 τ (B. rom Eq. (B., it can be seen that the parameter Z involves the rotation rate N and the coefficient of friction loss c f. The rotation rate N characterizes the effect of rotation on the axial mean velocity in the rotating pipe. The additional term c f /8 takes into accont the variation in the shape of the axial velocity profile at the pipe wall and, therefore, incldes the effect of different pressre losses in the pipe section de to rotation. igre B. shows the fnctional relationship between Z and Re D for varios N. It can be seen that for a given vale of N, the qantity Z increases with increasing vales of the flow-rate Reynolds nmber. This is cased by the decreasing vale of the friction factor with growing vales of Re D.

18 Appendix B: The lly Developed Velocity Profile in an Axially Rotating Pipe ig. B.: Rotation parameter Z as a fnction of the flow-rate Reynolds nmber with N as parameter (Weigand and Beer (994 If we assme in Eq. (B.5 to be far away from the wall so that /Re τ /dr/ is sfficiently small to be neglected in comparison with all other terms in this eqation, we can hope to find an niversal velocity law for the core region if the mixing length is only a fnction of r and Z. rom the above assmption, it is clear that the following analysis is only valid if N does not approach infinity. In this latter case, the whole pipe cross section is inflenced by the viscos forces and the axial velocity distribtion tends to the one for laminar pipe flow (Hagen-Poiseille flow = ( (B. This behavior is cased by the vanishing trblent shear stress with increasing rotation rate N. If we now exclde extremely large vales of N and introdce the dimensionless qantities defined in Eq. (B.6, we obtain from Eq. (3.9 for the Richardson nmber, Ri = 6rZ r (B. and the mixing length distribtion is given by Eq. (3.8. Becase the mixing length only depends on the radial coordinate and the Richardson nmber, it follows from Eq. (B.5 that the velocity profile in the core region mst have an niversal character. igre B.3 shows the distribtion of the axial velocity for varios vales of Z. It can be seen that the velocity profiles are

19 Appendix B: The flly developed Velocity Profile in an axially rotating Pipe ig. B.3: Axial velocity distribtion for varios vales of the rotation rate Z and different Re D (Weigand and Beer (994 niversal. There is only a difference in the profiles for different Reynolds nmbers near the wall ( r, where the viscos forces play an important role. In order to derive an approximation for the axial velocity distribtion, we follow the work of Reichardt (95 for the non-rotating pipe and sbdivide the flow area into two parts: a near wall region and a part which contains the rest of the flow area. Reichardt (95 developed, for the trblent flow in a nonrotating pipe, an approximation for the velocity distribtion which is valid for the whole flow area as well. This is also the aim for the case of trblent flow in an axially rotating pipe. or more details on this sbect, the reader is referred to Weigand and Beer (994. The Velocity Distribtion in the Core Region: If one plots the velocity distribtion of ig. B.3 in a diagram containing logarithmic axis, one obtains for the velocity distribtion C τ = Ar B (B.3 where A and B are depending on Z A( = B exp( ep( , (B.4

20 Appendix B: The lly Developed Velocity Profile in an Axially Rotating Pipe ig. B.4: Comparison between nmerical soltion and the approximation for the axial velocity in the core region (Weigand and Beer (994 or Z = (non-rotating pipe, the velocity distribtion given by the Eqs. (B.3- B.4 is identical to the one reported by Darcy (858. igre B.4 shows a comparison between the nmerically predicted axial velocity distribtion and the approximation given by Eqs. (B.3-B.4. It can be seen that Eqs. (B.3-B.4 approximate the velocity profile very well for.7. or larger vales of r, larger deviations can be noticed. The Velocity Distribtion in the Near Wall Region: It is well known that the velocity distribtion near the wall in a non-rotating pipe has a logarithmic distribtion given by y ln c κ (A.36 where κ and c are constants and y is the dimensionless distance from the wall, given by Eq. (A.3. In a rotating pipe, the trblent flctations are sppressed by the centrifgal forces. rthermore, the area, where the velocity profile is still logarithmic in shape, decreases with increasing rotation rates. This fact is shown in ig. B.5 and has been observed in DNS calclations by Eggels and Niewstadt (993 and by Orlandi and atica (995. However, Weigand and Beer (994 obtained an approximation formla for the velocity distribtion in the near wall region, which incldes the effect of relaminarization de to system rotation.

21 Appendix B: The flly developed Velocity Profile in an axially rotating Pipe 3 ig. B.5:Axial velocity distribtion / τ as a fnction of the wall coordinate y for two different vales of Z (Weigand and Beer (994 This eqation is given by (for more details the reader is referred to Weigand and Beer (994 Re ln ( 7.8 τ y y κ ep exp p exp p( ( y y with the qantities y y Re τ, (B.5 Re κ = κ τ Z /,.4 Re τ 3.5 (B.6 b = 3 κ κ In the eqations above denotes the shear-stress Reynolds nmber for the non-rotating pipe. As it can be seen from Eq. (B.5, the velocity distribtion in the near wall region is inflenced by a term containing the inflence of the system rotation on the wall shear stress. igre B.6 shows the good agreement between the approximation given above and the nmerical soltion of the problem.

22 4 Appendix B: The lly Developed Velocity Profile in an Axially Rotating Pipe ig. B.6: Comparison between nmerical soltion and the approximation for / τ in the near wall region (Weigand and Beer (994 An Approximation ormla for the Whole Region of the Pipe: Strictly speaking, the two eqations given above for the axial velocity distribtion are only valid in the near wall region and in the core region. However, the two soltions can be combined in one single relation, as shown by Weigand and Beer (994. They obtained Re ln τ y y exp ep( exp p κ y y { ( } C Ar B { ( } τ (B.7 with a = 5 ( Re /Re. rom the continity eqation in integral form, one can finally obtain a fnctional dependence between the shear-stress velocity and the τ maximm velocity in the pipe center C τ = Re D A Re B τ (B.8

23 Appendix B: The flly developed Velocity Profile in an axially rotating Pipe 5 ig. B.7: Inflence of a variation of the rotation parameter Z on the shape of the axial velocity profile (Weigand and Beer (994 ig. B.7 elcidates the inflence of the rotation parameter Z on the axial velocity distribtion for two different flow-rate Reynolds nmbers. It can be seen that increasing Z tends to relaminarize the flow. The axial velocity profiles tend to approach the parabolic distribtion of the Hagen-Poiseille flow for growing vales of Z. rthermore, it can be seen that the above obtained eqation for the axial velocity approximates very well the nmerical calclation. If the approximation is sed in ig.3.5 instead of the nmerical soltion, no noticeable difference can be seen. inally, it shold be noted here that the effect of pipe rotation is very mch different if a laminar flow enters the axially rotating pipe. In contrast to the previos discssions, the rotation cases for a laminar inlet flow a destabilization of the flow, even so the flow rate Reynolds nmber is mch smaller than. Mackrodt (97 fond that a laminar pipe flow is only stable against circmferential distrbances if the Reynolds nmbers are below the limits Reϕ 53.9 and ReD The reader is referred to the papers of Mrakami and Kikyama (98, Kikyama et al. (983, Reich et al. (989 and Weigand and Beer (99b for a more detailed discssion of this sbect.

24 Appendix C: A Nmerical Soltion Method for Eigenvale Problems Dring the analytical soltion process for the thermal entrance problems in pipe and channel flows, a linear partial differential eqation of the kind ( Θ Θ = x n r a n (3.9 has to be solved. If we consider a step change in the wall temperatre, the bondary conditions for this partial differential eqation are given by x = : (, n = : Θ n n= n = : (, (3.7 The soltion of the above given problem has been derived in Chap. 3 and is given by Θ = A exp exp(( = (3.55 In the following appendix, one possible nmerical soltion procedre for the eigenvale problems discssed in Chap. 3 and Chap. 4 will be presented. As it has been demonstrated, the soltion of the energy eqation can be obtained for these problems as a sm of eigenfnctions. In order to determine the temperatre distribtion, the eigenfnctions and the related eigenvales have to be predicted. or the above given problem (Eqs. (3.7, 3.9 the eigenfnctions are the soltions of the following ordinary differential eqation with the bondary conditions r ( ( ( (3.5 (3.6

25 8 Appendix C: A Nmerical Soltion Method for Eigenvale Problems The flow index in Eq. (3.5 has to be set to one for the heat transfer in pipe flows and to zero for the heat transfer in a parallel plate channel. Eigenvale problems, like the one given by the Eqs. ( , can normally only be solved nmerically, becase the coefficients in the differential eqation are complicated fnctions of the coordinate n. This is the case for the velocity distribtion and also for the fnction Pr = ε m Pr t a ( (C. One efficient method to solve these eigenvale problems is by sing a Rnge- Ktta method. This approach will be explained now in detail: In order to solve the Eqs. ( by a Rnge-Ktta method, we transform the second-order ordinary differential eqation into a system of two first-order ordinary differential eqation. Introdcing the new fnction χ = r a into Eq. (3.5 reslts in the following system of ordinary differential eqations χ ( /( = = χ r χ r (C. (C.3 In order to solve the system of eqations given by Eq. (C.3 with a Rnge- Ktta method, the problem has to be considered as an initial vale problem. Therefore, we provide an additional normalizing condition for the eigenfnctions according to ( (C.4 Using an arbitrary normalizing condition does not change the soltion of the problem, becase in Eq. (3.55 all the individal eigenfnctions are mltiplied by constants. This means that the vale prescribed by Eq. (C.4 will only inflence the vale of the constants A in Eq. (3.55. The algorithm for the Rnge-Ktta method for solving a system of ordinary differential eqations is relatively simple (see for example Törnig (979. The system of differential eqations, given by Eq. (C.3, is now solved together with the two initial conditions ( ( χ (C.5 This will be done by sing a gessed vale of the eigenvale λ a. Examining the bondary condition for the eigenfnction for ( ( shows, if the gessed vale λ a is an eigenvale of the problem nder consideration. If λ a is not an eigenvale, the calclation procedre will be contined with a different

26 Appendix C: A nmerical Soltion Method for Eigenvale Problems 9 gessed vale λ b λ a. This procedre, which is known in literatre as a shooting method, will be contined p to the point when a change in sign appears for (. If this is the case, an eigenvale has been fond in the interval λ a, λ b. This eigenvale can then be predicted to whatever accracy is desired, by sccessively halving the interval. This means that the eigenvale can finally be predicted with the accracy ( ε (C.6 6 =. The here discssed proce- where it is normally accrate enogh to set dre is illstrated in ig. C.. ε ig. C.: Nmerical prediction of the eigenvales The nmber of zero-points of the eigenfnctions increases linearly with growing vales of. This means that higher eigenfnctions strongly oscillate and that it is difficlt to captre exactly the shape of the eigenfnctions for larger by a nmerical method. In Chap. 4, we have investigated an analytical methods for predicting sch eigenfnctions for large vales of. However, if the nmber of grid points sed to resolve the interval [ in the nmerical prediction is large ] enogh, also higher eigenfnctions and eigenvales can be predicted accrately. or this normally grid points have shown to be sfficient. C. Nmerical Tools On the internet web page: the reader will find a page entitled Analytical Methods. The reader can get the login and the password for this page pon reqest from the athor (bw@itlr.ni-stttgart.de. Here are several sefl programs (exectables and sorce codes for the prediction and animation of thermal entrance heat transfer problems. The programs for predicting thermal entrance problems in a parallel plate channel will be explained here in a little more detail, becase for these programs also the sorce code is provided on the web page stated above.

27 3 Appendix C: A Nmerical Soltion Method for Eigenvale Problems Veldch This program (Velocity-Profile-D-Channel, which is written in ORTRAN, predicts the hydrodynamically flly developed velocity profile for a laminar or trblent channel flow. The nderlying eqations, which are solved, are discssed in detail in Appendix A. At the start of the program the reader is asked to provide a Reynolds nmber for the channel flow. If a vale for the Reynolds nmber, based on the hydralic diameter is provided, which is smaller than, the program atomatically calclates the velocity profile for a laminar channel flow (see Eq. (3.. In addition to the soltion of the hydrodynamically flly developed velocity profile, the program also predicts the fnction given by Eq. (C. and the flly developed temperatre distribtion in case of a constant wall heat flx. or the trblent heat diffsivity, the trblent Prandtl nmber model by Weigand et al. (997a is sed. or laminar flow, the fnction a, and the flly developed temperatre distribtion is given by Eq. (5.76 (for Pe L. This program serves as an inpt generator to the program Tempdch (Temperatre-Distribtion-D- Channel, which solves the energy eqation for the thermal entrance problem in a parallel plate channel. Tempdch This program, which is also written in ORTRAN, predicts the temperatre distribtion and the Nsselt nmber for a hydrodynamically flly developed flow. The nderlying eqations are discssed in detail in Chap. 3. Soltions of the energy eqation are provided for the case of constant wall temperatre and for the case of constant wall heat flx. The program ses the velocity distribtion, the fnction a and the flly developed temperatre profile (which is only needed in case of a constant wall heat flx bondary condition from the program Veldch. The calclation of the eigenvales is done by sing a Rnge-Ktta method and follows the approach otlined in the first part of this chapter. Visalization of Reslts After execting the above described programs, several data files are generated, which can be visalized by freeware programs as explained on the previos mentioned web page. The otpt files contain all main interesting calclated qantities like temperatre profiles (line plots and the development of the profiles as a fnction of axial coordinate, Nsselt nmber distribtion, etc.. All the programs provided on the internet page have been rewritten from older ORTRAN programs in order to be more ser friendly. The athor has sed similar programs for a long time and a lot of comparisons have been made over time between predicted eigenvales, constants, Nsselt nmber distribtions and literatre data. This incldes comparisons for pipe flow with Shah (975, Shah and London (978, Papotsakis et al. (98a (with axial heat condction, compari-

28 Appendix C: A nmerical Soltion Method for Eigenvale Problems 3 sons for parallel plate channels with Brown (96, Deavors (974, Shah and London (978 and for the case of circlar annli with Lndberg et al. (963 and with Hs (97 (with axial heat condction. Table C. shows a comparison of present predictions with vales reported by Shah (975 for a laminar flow in a parallel plate channel with constant wall temperatre. It can be seen that the present calclations are in excellent agreement with the vales of Shah (975. This is interesting, becase Shah (975 obtained 4 for x /( Pe D the vales for the Nsselt nmber from an extended Leveqe soltion. Table C.: Comparison of local Nsselt nmbers for the thermal entrance region in a parallel plate channel with laminar internal flow and constant wall temperatre x /( Pe D N D (Shah (975 N D (own calclation In addition, this sort of programs have been compared against nmerical predictions for pipe flows (for example Hennecke (968 and for circlar annli (ller and Samels (97. Some of the above mentioned comparisons have already been shown in Chap. 3 and Chap. 4. The reader will find it relatively straight forward to extend the programs provided on the above given web page to The series soltion for the temperatre field and for the Nsselt nmber, which have been obtained in Chap. 3, are converging very slowly for x. Leveqe (98 developed an approximation for the temperatre field for very small vales of x. He approximated the velocity profile, within the very thin thermal bondary layer by n. By doing so, the energy eqation has a similarity soltion (see Chap. 6. This soltion can be sed 3 as long as the thermal bondary layer thickness is very thin ( x /( Pe D. The reader is also referred to Worsoe-Schmitt (967 and to Kader (97, where also soltions of this kind are reported for the extended Graetz problem (with axial heat condction.

29 3 Appendix C: A Nmerical Soltion Method for Eigenvale Problems the case of pipe flow and flow in circlar annli. Also the extension to solve eigenvale problems for sitations, where the axial heat condction cannot be ignored, is relatively easy. One Example On the web-page the reader will find also some example calclations and a docmentation in order to show how to se the programs. In addition also some comparisons between the programs and literatre data are provided. One simple example, which cold be sed as a start, in order to get sed to the programs, cold be to compte the laminar heat transfer for a slg-flow velocity profile. Here one can develop very easily a complete analytical soltion. Let s assme that ( (C.7 This cold be the case for a laminar flow of a liqid metal with a very low Prandtl nmber. Inserting Eq. (C.7 into the energy eqation (3.5 reslts in Θ = x y Θ (C.8 Let s now consider constant wall temperatre bondary conditions according to Eq. (3.7. Then we obtain for the planar channel x : (, (, y = : Θ y y= y : (, (3.7 The soltion of Eq. (C.8 together with the bondary conditions given by Eq. (3.7 can be easily obtained by the method of separation of constants and one gets Θ=A = cos( exp( (C.9 with the qantities A sin( π, λ,,,,... (C. The niform velocity profile given by Eq. (C.7 can easily been modified in the inpt file for the program Tempdch by setting all vales for the velocity profile to one. A comparison between the analytical predicted eigenvales and the nmerical calclated vales is reported in Table C. It can be seen that there is an excellent agreement between the nmerically predicted vales and the exact soltion.

30 Appendix C: A nmerical Soltion Method for Eigenvale Problems 33 Table C.: Comparison between analytically predicted eigenvales and calclated eigenvales by sing the program Tempdch Nr. ( analytical ( nmerical Rel. Error < < < < < < < < < -7 The same sort of accracy can also be achieved for the constants A. A comparison between nmerically predicted vales and the exact soltion is given in Table C.3 Table C.3: Comparison between analytically predicted constants A and nmerically calclated vales by sing the program Tempdch Nr. ( analytical ( nmerical Rel. Error < < < < < < < < < 4-6 More examples can be fond on the previos mentioned web-page. Please note that the provided programs on the web-page are for edcational se only. They shold not be sed for any commercial applications. rthermore, the athor does not take any warranty for any reslts prodced by the programs.

31 Appendix D: Detailed Derivation of Certain Properties of the Method for Solving the Extended Graetz Problems In the following appendix, certain properties of the method explained in Chap. 5 are derived in detail. Where necessary, the derivation is shown for the two different bondary conditions considered in Chap. 5: constant wall temperatre and constant wall heat flx. D. Symmetry of the Matrix Operator L We start or considerations by showing that the matrix operator L is a symmetric operator in the Hilbert space H = H H, where H is the space containing all fnctions f in [ ] for which the integral f dn < holds. H is the space of all fnctions b( in [ ] for which Eq. (D. holds b ( r a ( dn < (D. (D. Now it can be shown, that the matrix operator L, given by Eq. (5., is symmetric with respect to the inner prodct defined by Eq. (5.4. We assme that the two vectors, ϒ D, which is defined for a given wall temperatre belong to by Eq. (5.5 and for a given wall heat flx by Eq. (5.59. This means that we have to show, that, L L, ϒ (5.6 Eqation (5.6 can be proven by inserting the expressions L and Lϒ into the definition of the inner prodct according to Eq. (5.4. This reslts in

32 36 Appendix D: Detailed Derivation of Properties for the Extended Graetz Problem r, Lϒ = ( dn Pe L (5.7 r L,, ( n ( n ( n ( n ( n ( n dn Pe L (5.8 Sbtracting Eq. (5.8 from Eq. (5.7 reslts after integration in,,, ( ( (5.9 ( ( ( ( ( The reslting expression on the right hand side of Eq. (5.9 is zero for the case of a given wall temperatre becase ( (, as well as for the case of a given wall heat flx bondary condition, becase ( (. or both bondary conditions = ϒ ( = becase of the assmed symmetry of the eigenfnctions. D. The Eigenfnctions Constitte a Set of Orthogonal nctions Next, we show that the eigenfnctions given in Chap. 5 constitte a set of orthogonal fnctions. Consider the two eigenfnctions and k together with the related eigenvales λ and λ k. rom Eq. (5. one obtains: L = λ (D.3 L k = λ k k with the bondary conditions Constant Wall Temperatre k ( k ( (D.4 Constant Wall Heat lx k k ( (D.5 By taking the inner prodct (defined by Eq. (5.4 of Eq. (D.3, one obtains (note, that from Eq. (5.4 follows that, =,, k k

33 Appendix D: Detailed Derivation of Properties for the extended Graetz Problem 37 L,, k L k =(, k (D.6 In the last paragraph, it has been shown that the operator L is a symmetric operator in the Hilpert space. This means that Eq. (5.6 is valid, L L, ϒ (5.6 where and ϒ can be replaced by and k. Then it follows immediately from Eq. (D.6 that, k rom this eqation, one can conclde that for λ, k = for λ λ λ k k (D.7 (D.8 Eq. (D.8 shows that the eigenfnctions constitte a set of orthogonal fnctions for the inner prodct defined according to Eq. (5.4. D.3 A detailed Derivation of Eq. (5.3 and Eq. (5.6 Becase of the non-homogeneos bondary conditions, the vector S does not be- given by Eq. (5.5 (for constant wall temperatre long to the domain D bondary conditions or by Eq. (5.59 (for constant wall heat flx bondary conditions. However, it can be shown that Eq. (5.3 and Eq. (5.6 hold. L S,, ( (5.3 L S,,, ( (5.6 These two eqations can be derived in the following way: from the definition of the inner prodct by Eq. (5.4 one obtains LS, r Pe L ( ns ( n ( n S ( n (n S ( n dn (D.9 r S,, n n ( ns n ns n dn Pe S L (D.

34 38 Appendix D: Detailed Derivation of Properties for the Extended Graetz Problem where the prime denotes the derivative with respect to n. rom these two eqations one obtains LS,,, ( ( (D. ( ( where the soltion vector S was defined in Eq. (5. by Θ (, S = E E( (, ( (D. Therefore, Eq. (D. can also be written as L S,,, ( ( (D.3 ( ( ( or the case of a constant wall temperatre, the bondary conditions are given by Θ n n = =,,, given In addition, it follows from Eq. (5. that ( E (D.4 and from the definition of the fnction E by Eq. (5.9 it follows that (. Therefore, Eq. (D.3 simplifies for the case of a given wall temperatre to L S,,, ( (D.5 If now Θ is replaced by the bondary condition according to Eq. (5.7, Eq. (5.3 is obtained. or the case of a given wall heat flx, the bondary conditions are given by ( ( Θ n n n, Θ =, = given Remembering that ( and E considering that ( (D.6, one obtains from Eq. (D.3, by for the case of a prescribed wall heat flx, Eq. (5.6 LS,,, ( ( (5.6

35 Appendix D: Detailed Derivation of Properties for the extended Graetz Problem 39 D.4 Simplification of the Expression for the Temperatre Distribtion (for Constant Wall Temperatre In Eq. (5.4 we sed the reslt that ( ( ( λ λ = (5.4 This expression can be derived in the following way. We start by expanding a vector f = T according to Eq. (5.9. This reslts in f, (D.7 = = The expression f, can now be evalated from Eq. (5.4 which reslts in a f, ( Pe L r ( n dn rom this eqation, one obtains for the first vector component of Eq. (D.7 = a r = Pe L dn (D.8 (D.9 Now the integral in Eq. (D.9 can be rewritten. irst, one can replace the integrand by sing Eq. (5.. This reslts in a n r dn PeL λ r dn (D. ( The integral on the right hand side of Eq. (D. can be expressed as n r dn r dn λ λ r dnˆ ˆ dn λ (D. Evalating the first expression on the right hand side of Eq. (D. shows that this term is zero. If one frther replaces by Eq. (5., one obtains n n dn λ r dn ˆ dn = r dnˆ r a ˆ If one replaces now the integral in Eq. (D.9 one obtains γ n dn (D.

36 4 Appendix D: Detailed Derivation of Properties for the Extended Graetz Problem = γ ( dn (D.3 In order to show that this expression is identical to Eq. (5.4, one has to show that the first sm on the right hand side of this eqation is zero. Expanding the vector T f = into a series, one obtains for the first vector component = γ d = dn (D.4 Eq. (D.4 shows that the first sm in Eq. (D.3 is zero and one obtains therefore = = (D.5 If we distingish explicitly between positive and negative eigenfnctions in Eq. (D.5, we obtain Eq. (5.4. D.5 Simplification of the Expression for the Temperatre Distribtion (for Constant Wall Heat lx or the case of a prescribed wall heat flx, we can simplify Eq. (5.66 by replacing the first two terms in this eqation. In the following section, it will be explained how this can be done. The first term in Eq. (5.66 to be replaced is = λ ( We start or considerations by expanding the vector f = T the first vector component one obtains = a r Pe = L dn (D.6 into a series. or (D.9 The integral in the above eqation can be rewritten by sing Eq. (5.3. This reslts in a r dn = Pe L λ d dn r a n dn λ r dn (D.7

37 Appendix D: Detailed Derivation of Properties for the extended Graetz Problem 4 Carrying ot the integrations in the above eqation, it can be seen that the first integral is zero for the case of a prescribed heat flx at the wall and one obtains = r d λ = dn (D.8 The integral in Eq. (D.8 can be frther rewritten by partial integration. This reslts in ( ( ( (n dn λ i i λ ω( r dn r a n (D.9 In order to obtain an expression for Eq. (D.6, the second term on the right hand side of Eq. (D.9 needs to be evalated. Expanding the vector = into a series reslts in f ( T f, = ( ω = ω ( ( d dn (D.3 Taking the first vector component of Eq. (D.3 shows that the second term on the right hand side of Eq. (D.9 is zero and we obtain for the expression given by Eq. (D.6 = λ ( = (D.3 In addition, we want to derive an analytical expression for the first sm on the right hand side of Eq. (5.66: = λ ( (D.3 If we pt x in Eq. (5.66, we obtain the flly developed temperatre distribtion in the dct with a prescribed heat flx at the wall. Using Eq. (D.3 this distribtion can be written as ( x : Θ ( xn ( (D.33 = λ

38 4 Appendix D: Detailed Derivation of Properties for the Extended Graetz Problem As it can be seen from Eq. (D.33 this temperatre distribtion contains a term, which is linearly increasing in x and a term, which is a fnction of n. Therefore, Eq. (D.33 might be written as x : (D.34 The fnction Ψ, appearing in Eq. (D.34, is not known. However, this (5.6 and solving the reslting ordinary differential eqation for fnction can easily been obtained by inserting Eq. (D.34 into the energy eqation Ψ. One finally gets n n ( r dsdn C r a ( dd C Ψ = (D.35 The constant C, which appears in the above eqation, can be derived from a global energy balance. This might be done by evalating Eq. (5.9 for the case of a thermally flly developed flow for the whole flow domain E. By considering the bondary conditions for the fnction E, given by Eq. (5.58, one obtains x : x a r dn Pe L x (D.36 Inserting the temperatre distribtion given by Eq. (D.34 into Eq. (D.36 reslts in dn n Pe L a r dn (D.37 Comparing Eq. (D.37 with Eq. (D.35 reslts in the following expression for the constant C = a C r dn Pe L n n ( Ψ = r ds dn r a ( rom this eqation one finally obtains for the sm, Eq. (D.3, ( Ψ ( n C = λ (D.38 (D.39 (D.4

39 Appendix D: Detailed Derivation of Properties for the extended Graetz Problem 43 It shold be noted here that the above analytical expressions, Eqs. (D.6 and (D.3, transform into the expressions developed by Papotsakis et al (98, if we consider the simplified case of a laminar pipe flow ( =, a = a =,. or this case one obtains = λ = (D.4 and = ( ( λ 4 n 8 7 = 4 4 n e D (D.4 These expressions have also been given in Eq. (5.75. or the case of a planar channel, Eq. (5.76 is obtained. D.6 The Vector Norm In this section, some interesting reslts abot the vector norm are presented. The vector norm is defined by inserting the eigenvector into Eq. (5.4. This reslts in =, a r ( r dn n ( Pe L a r (D.43 irst, we establish a connection between the vector norm and the derivative of the eigenfnction with respect to the eigenvale (see Eq. (5.46 and Eq. (5.7. This connection is important becase it allows s to evalate the constants in the temperatre distribtion very easily. Therefore, we consider two vectors and which both satisfy Eq. (5.. This reslts in L = λ L= λ The eigenvector (D.44 satisfies both bondary conditions at and, whereas the vector satisfies the bondary condition at and only in the limit λ λ the bondary condition at. If we now apply the inner prodct, given by Eq. (5. 4, to Eq. (D.44, we obtain