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and in an axisymmetric combustion chamber (Fig. any physics missing in the k-equation causes the loss of accuracy in describing the second scale transport regardless the choice for this scale. II. the “rapid” pressure-diffusion term does not appear in equation (1) and requires modeling. Therefore. Evidence that important information is missing in equation (1) is that two-equation models employing this equation can generally reproduce the mean velocity and shear stress profiles accurately. The contribution of the pressure diffusion to the transport equation for ε ( ε -equation) appears through one of the modeling coefficients in the ε -equation. Poroseva and Bézard9) where wall effects are irrelevant.model ∂ (ν t σ k ⋅ ∂k ∂xi ) ∂xi for the turbulent diffusion in equation (1) can absorb only a part of the pressure-diffusion term < ui ∂p ∂xi > . the so-called “slow” part: − 1 ρ < ui ∂p ( s ) 1 ∂ < umumui > > = ∂xi ∂xi 5 (2) (Lumley7). even in the central core of free shear flows (Poroseva8. but the turbulent kinetic energy level is not well predicted. the pressure diffusion correlation in the exact transport equation for the turbulent kinetic energy can 3 . MODELING THE PRESSURE DIFFUSION Excluding from the consideration a flow area very close to the flow boundary. in a channel with wavy walls. A two-equation model with the new k-equation was validated in Poroseva8 in self-similar free shear flows (plane wake. Notice also that the transport equation for the dissipation ε (or any other second scale) is formally derived from the k-equation. ρ is the density and ∂ < umumui > ∂xi is the turbulent diffusion. The objective of the current paper is to investigate further potential of the model in application to simulations of the separated flows in a planar diffuser. plane and round jets) and equilibrium boundary layers with and without pressure gradients. Thus. It was shown by Poroseva8 that the turbulent kinetic energy level could be corrected if the “rapid” part of the pressure diffusion term (related to the mean velocity gradients) is included in the k-equation as an extra term related to the production P. over a backstep. In (2). 1). plane mixing layer.
The velocity – pressure-gradient correlation on the left side of (5) is evaluated at point Y. its contribution in the turbulent kinetic energy balance is absorbed by an adopted model for the turbulent diffusion. A model for this term 1 ∂ < ui p >( r ) ⎛ 3 ⎞ = ⎜ − + Ck ⎟ P . The first term on the right hand side of (3) is called the “rapid” part due to its relation to the mean velocity gradient.n . which ranges over the region of the flow. ′ ∂xn ∂xm ⎦ r ⎣ − 1 ρ < ui ∂p ( r ) 1 > = − ∂x j 2π ∂ ∫∫∫ ∂x′j (5) In (5). The second term on the right hand side of (3) can be modeled by expression (2) as discussed above. dV' is the volume element. r is the distance from Y' to the point Y with coordinates xi . “ ’ ” above a flow variable indicates that it should be evaluated at a point Y' with coordinates x'i . (6) 4 . j >( r ) = anmjiU m. Expression (4) is obtained by analyzing the properties of the exact integral expression for the “rapid” part of the velocity – pressure-gradient correlation in an incompressible flow ⎡ ∂U m ∂ < un ui > ⎤ 1 ′ ′ ⎢ ′ ⎥ dV ′ .be presented as a sum of “rapid” and “slow” parts 1 ∂ < ui p > 1 ∂ < ui p > ( r ) ∂ < ui p > ( s ) ) =− ( + ∂xi ∂xi ∂xi ρ ρ − (3) (Chou10). ρ ∂xi ⎝ 5 ⎠ − (4) was first suggested in Poroseva8. Therefore. whereas all derivatives on the right side are taken at Y' . The analysis yields the following model for this correlation − where 1 ρ < ui p.
anmji = − 1 4 ( < uiu j > δ mn + < uium > δ jn ) + 5 < uiun > δ jm + 5 ⎡1 C1 ⎢ ( < ui u j > δ mn + < ui um > δ jn ) + k (δ ijδ mn + δ imδ jn ) + < ui un > δ jm ⎣2 − < u j um > δ in − 2 ( < u j un > δ im + < umun > δ ij ) ⎤ + ⎦ (7) ⎡1 C2 ⎢ ( < ui u j > δ mn + < ui um > δ jn − < u j un > δ im − < umun > δ ij ) + ⎣2 3 ⎤ + kδ inδ jm − < u j um > δ in ⎥ . in homogeneous turbulence. the coefficients C1 and C2 are linked: 1 5 − C1 − C2 = 0 5 2 (9) (Poroseva11). In homogeneous turbulence. 2 ⎦ In expression (7). Notice that these expressions are derived without the assumption of homogeneity. 2 (8) In general. Derivation of expressions (6) and (7) is described in detail in Poroseva11. δ ij is the Kronecker symbol. the pressure diffusion term does not contribute to the turbulent kinetic energy balance as expected. Substitution of (9) in expression (8) yields the universal constant value of Ck equal to 0. expression (6) with model (7) for the tensor function anmji contracts to (4). however. the coefficient Ck is a function of the same parameters as the coefficients C1 and C2 . where Ck = 15 ⋅ C1 + 3C2 . In the transport equation for the turbulent kinetic energy. the model coefficients C1 and C2 are generally unknown functions of several parameters. It should be pointed out that model expression (4) for the “rapid” part of the pressure 5 .6 . That is.
not necessarily of the diffusive type. C2 . and < u3 p > . ⎟ σ k ⎠ ∂xi ⎥ ⎢⎝ ⎣ ⎦ Dk ∂ = ( 0.3 vanishes throughout the flow.6 . Notice.2 .i is whether a model expression for this term should be of the diffusive type. however.12 by analyzing direct numerical simulation (DNS) data. and Ck . Finally. one obtains a new model equation for the turbulent kinetic energy transport: ⎡⎛ ν t ⎞ ∂k ⎤ ⎢⎜ν + ⎥. III. (More discussion on a functional form of the model coefficients is provided in Section IV.1 . This requirement does not also imply that each of the terms in expression (3) (which in its complete form includes boundary effects) would vanish separately. even assuming that the integral of (4) taken over the entire flow volume should vanish. TURBULENCE MODEL Using model expression (4) for the “rapid” part of the pressure diffusion.4 + Ck ) P − ε + Dt ∂xi (10) In homogeneous turbulence with Ck = 0. In regard to the “rapid” part of the correlation < ui p. The form of equation (10) does not require any modification in the standard ε -equation 6 . < u2 p > . It does not imply that the sum of three correlations < ui p > .3 ) taken over the entire flow volume vanishes. one can argue that this result can be achieved with different functional forms of the coefficient Ck .i vanishes at every point in the flow. or that any one of < u1 p > . 2. that expressions (4) and (7) hold regardless of the models for the coefficients C1 .i ( i = 1.) This question clearly requires more study in the future. equation (10) transforms to the standard k-equation (equation (1)). An important question in modeling < ui p > . j > (see expression (5)).diffusion is similar to the model derived ad hoc by Demuren et al. What “diffusive type” requires is that the integral of the sum of three correlations < ui p > . there is no indication that the model for this term should be of the diffusive type.
It is interesting to note that the same value Cε 1 = 1. Cε 2 . one can reproduce well not only the mean velocity and shear stress profiles. However.67 = 1. Cε 2 = 1.92. the turbulent kinetic energy level is reproduced correctly with Cε 1 = 1.5 was recommended as optimal for homogeneous flows also (Kassinos et al. ⎟ σ ε ⎠ ∂xi ⎥ ⎢⎝ ⎣ ⎦ (11) where Cε 1 .5 in Poroseva and Bézard9). boundary layers). mixing layer. and the mixing layer. 7 . if one allows the coefficient Cε 1 being constant in any given flow. since a functional form is currently unavailable for both coefficients.14) models. However.5 (see Fig.15). The results of simulations obtained with Cε 1 variable from flow to flow (see Table 2 in Poroseva and Bézard9) and the rest of the coefficients given by ν t = Cμ k 2 ε . and σ ε are model coefficients.09. the plane jet. The effect of the “rapid” part of the pressure diffusion comes from the coefficient Cε 1 . Only in the round jet.Dε ε ∂ = ( Cε 1 P − Cε 2 ε ) + Dt k ∂xi ⎡⎛ ν t ⎞ ∂ε ⎤ ⎢⎜ν + ⎥.6 and both coefficients Ck and Cε 1 vary from flow to flow as shown in Table 1. the axis level of the turbulent kinetic energy is either overestimated (plane wake) or underestimated (plane jet. Cμ = 0. but also the turbulent kinetic energy level in the plane wake. we approximate them in this study by constant values and then. but variable from flow to flow. as well as in the equilibrium boundary layers (Poroseva8). Indeed. investigate whether this approximation results in improvement of simulation results. even the standard k − ε model (equations (1) and (11)) reproduces the mean velocity and shear stress profiles in a good agreement with experimental data in free shear flows and equilibrium boundary layers under different pressure gradients (Poroseva and Bézard9).5 (12) are better than the results obtained with the k-ω (Wilcox13) and k-ϕ (Cousteix et al. Allowing the coefficient Ck deviate from its homogeneous value 0. which generally should be a function of the same parameters as the coefficient Ck . σ ε σ k = 1 / 0.
Steady equations are solved using the commercial code Fluent. 2(a)). IV. Figure 2 shows the experimental and calculated profiles of the streamwise velocity and the turbulent kinetic energy (both scaled by the inlet velocity) at three spatial locations in the streamwise direction x. in a channel with wavy walls.44 . On this and other figures. Cμ = 0. and the 8 . modeling wall effects (or low Reynolds number effects) is not in the focus of the current paper. Their values are chosen such as to fit the experimental data. but variable from flow to flow.09 . and (11) are written in the high Reynolds number form. (10). the standard damping function approach proposed by Launder and Sharma2 is used to correct the behavior of turbulent quantities in the viscous dominated near-wall regions.3 . and σ ε σ k = 1. The coefficients Ck and Cε 1 for the model LS-RPD are given in Table 2. The Reynolds number based on the inlet velocity ( ui ) and the step height (H) is 5. Cε 2 = 1.3 / 1 = 1. dashed lines show the profiles calculated with the LS model. with the model coefficients being set to their standard values: Cε 1 = 1.1(a). The performance of the LS-RPD model will be compared with the performance of the standard k − ε model (Launder and Sharma2). 1). and in an axisymmetric combustion chamber (Fig. over a backstep. This model (hereinafter referred as LS) includes equations (1) and (11).100. Results obtained using the four-equation < v 2 > − f model (Durbin16) are also included for comparison. Separation is fixed at the step and the expansion generates a large recirculating region with strong negative velocity and high turbulent kinetic energy (measurements are available at several stations downstream the step). The level of the turbulent kinetic energy at this location is overpredicted by the LSRPD model and underpredicted by the LS model. RESULTS AND DISCUSSION The first problem selected is the backstep flow (Jovic and Driver17) shown in Fig. The flow at the inlet is a fully developed boundary layer. the k − ε model including equations (10) and (11) with the coefficients Ck and Cε 1 being constant. We will denote this model as LS-RPD. y is the vertical direction.In Section IV. solid lines show the LS-RPD model profiles. Since. Equations (1). and < v 2 > − f – give the same mean velocity profile upstream ( x / H = −3 ) of the backstep (Fig.92 . and with the rest of the model coefficients given by (12) is validated in four separated flows: a planar diffuser. All three models – LS. LSRPD.
which is very challenging for turbulence models. three models produce reasonably accurate results at two spatial locations x / H = 0. A central pipe stream and an annular swirling stream enter a large cylindrical chamber. 1(b). The Reynolds number based on the bulk velocity ( ui ) and the inlet height (H) is 20. The fourth case consists of the axisymmetric combustion chamber (Fig. The flow separates on the downhill slope and reattaches on the uphill. which are comparable in accuracy for the mean velocity and the turbulent kinetic energy at all three locations. The presence of a mild adverse pressure gradient induces a separation on a smooth surface. The value of Cε 1 is chosen to fit the experimental data for the mean velocity (see Table 2)). As in the previous case.6. whereas other two models are in a good agreement with the experimental data. 1(c)).dash-double-dotted lines correspond to the results produced with the < v 2 > − f Experimental data are given by white circles. and in response to a 9 .75. Downstream. The flow is fully developed at the inlet. Since the level of the turbulent kinetic energy is not known in this case. 5).1(d)). 32 ).25 and 0. Mean velocity and turbulent kinetic energy profiles are available as well as wall skin friction (Buice and Eaton18) to identify the extent of the separated region. both models capture the extent of the separation region very well (Fig. model. whilst the LS-RPD and < v 2 > − f models produce similar friction levels (Fig. the LS results are in a poor agreement with the measurements (Fig. 28. These results are in a good agreement with the experimental data. the LS model does not correctly reproduce the separation zone at both locations ( x / H = 4 and 6). As Figure 6 demonstrates.000. In addition. The third case is the flow in a periodic wavy channel (Fig. 3) The second test case is the flow in the asymmetric diffuser shown in Fig.000. 4) at three streamwise locations ( x / H = 24 . Only velocity measurements are available in this case (Kuzan19). Both LS-RPD (with the coefficients Ck and Cε 1 shown in Table 2) and < v 2 > − f models produce results. its homogeneous value. the value of Ck cannot be chosen and is set to 0. Also. The Reynolds number based on the bulk velocity ( ui ) and the average channel height (H) is 11. the LS model fails to calculate the correct friction coefficient c f in the recirculating bubble and underestimates c f in the recovery region.
6 corresponding to homogeneous turbulence and Cε 1 chosen to fit the experimental mean velocity profile. that is.6 ) in the chamber (Hagiwara et al. whereas the mean velocity and shear stress profiles are universal (or nearly universal) under appropriate scaling. (The rest of coefficients is set to (12). However. however.) Interesting enough. the optimal value found for Cε 1 ( Cε 1 = 1. in the round jet and diffuser flows. The Reynolds number based on the pipe bulk velocity ( ui ) and diameter ( = 2 R ) is 75. the normal stresses and turbulent kinetic energy profiles are non-unique. Currently.68. a general functional form should be found for Ck and Cε 1 . The LS model considerably overestimates the extent of the recirculating bubble and reproduces poorly the streamwise and swirl velocities at the location x / R = 1.1. That is.20) Again. in simulation of flows for which experimental data are not available. multiple asymptotic states can be observed (see discussion in Rogers and Moser5 and Moser et al.6) That is. The value of Cε 1 is given in Table 2. the performance of three models is comparable in accuracy and is in good agreement with the experimental data. The results reported in this paper and Poroseva8 demonstrate that the approximation of the coefficients Ck and Cε 1 by constants works well in all test flows with different geometries and at different Reynolds numbers provided that the values of these coefficients are allowed to change from flow to flow. only few conclusions about the coefficients Ck and Cε 1 can be drawn based on available experimental and DNS data. to use equations (10) and (11) for predictions. they can be well reproduced with Ck = 0.5 ) is the same as recommended for homogeneous turbulence in Kassinos et al.15.68 (Fig. It appears. that in geometrically-equivalent flow situations at the same Reynolds number. What determines the final choice of the set ( Ck . a recirculating region is created. At other locations. Streamwise (u) and swirl velocities (w) are measured at three stations ( x / R = 0. 7 (b)). because experimental data is not available for the turbulent kinetic energy. Cε 1 ) in a given flow is the turbulent kinetic energy level controlled by the coefficient Ck .strong adverse pressure gradient. 3. DNS confirms that multiple asymptotic states reflect the differences in the large-scale structure of 10 . the mean velocity and shear stress profiles (if available) appear to be insensitive to the value of the coefficient Ck .7 . In each test flow. the coefficient Ck in the LS-RDT model is set (not chosen) to 0.6.000.
the contribution of the “rapid” part of the pressure diffusion to the turbulent kinetic energy balance was modeled.6 . which depends strongly on the Reynolds number. e. second derivatives of flow characteristics and other parameters not specified yet. This term contains the model coefficient Ck . The standard ε -equation does 11 . Nevertheless. V. which is generally a function of unknown parameters related to the large-scale structure of turbulence and inhomogeneous effects. that is. (Tsinober21). Thus. no single value can be recommended for all considered flows. C2 .g. The coefficient Ck is linked to the coefficients C1 and C2 through expression (8). and Ck . As for the coefficient Cε 1 .6). to the pressure diffusion effects. In the transport equation. external forces etc. boundary conditions. the coefficient takes the universal value: Ck = 0. functional forms for both coefficients cannot be considered separately. “uncontrolled and possibly unknown properties of the initial or inlet conditions” (Moser et al. which controls the level of the turbulent kinetic energy. More study (including DNS) is necessary to determine general functional forms for the coefficients C1 . In homogeneous turbulence. No universal constant values exist for C1 and C2 . Since this is the coefficient Ck . This fact clearly indicates that C1 and C2 should also depend on parameters which directly relate to inhomogeneous effects such as. However. Since variability of both coefficients Ck and Cε 1 is linked to the same mechanism. Ck should be a function of the same parameters as these two coefficients. A new k − ε model with the additional term linked to the pressure diffusion effects in the transport equation for the turbulent kinetic energy was derived. the coefficient Ck does have such a value in homogeneous turbulence: Ck = 0. one can show that C1 and C2 are functions at least of the mean velocity gradients and the Reynolds stresses even in homogeneous turbulence.turbulence. its value in different flows is close to each other (see Tables 1 and 2). the large-scale structure should be reflected through this coefficient.6 . Considering the limiting states of turbulence (Poroseva11). flow geometry. the effect of the “rapid” part of the pressure diffusion manifests itself through an additional term related to the production term.. SUMMARY In the current paper.
and the turbulent kinetic energy as well as the skin friction coefficient can be achieved if one sets these two coefficients to be constant in any given flow. Svetlana V. no two-equation model. The results obtained in these flows complement the results for free shear flows and equilibrium boundary layers reported previously in Poroseva8. the shear stress. “Consistency” of models for different terms means that all models are derived based on the same assumptions and approximations. The new model was validated in four separated flows: a planar diffuser. are not sufficient to understand and describe this connection between the model coefficients and the large-scale structure in a general form. in a channel with wavy walls. The first author would also like to thank Robert Rubinstein (NASA-Langley Research Center) and M. The proposed form of the k-equation is not complete in a sense that model expressions used in the equation to represent other terms are not consistent with each other and with the models for both parts of the pressure diffusion term. In this sense. The results of simulations of the separated flows with the new k − ε model are much better than the results produced with the standard k − ε model and comparable in accuracy with the computational results of more complex four-equation < v 2 > − f model. however. Poroseva conducted a part of this research when was affiliated with the Center for Turbulence Research (Stanford University). at least partially. they were approximated by constant values in simulations of all test flows. but variable from flow to flow. Variability in values of the coefficients can be linked. which is a counterpart of the coefficient Ck in the k-equation. and in an axisymmetric combustion chamber. Y. over a backstep. Available data. The pressure diffusion effects influence this equation through the coefficient Cε 1 . Florida State University) for support in the preparation of this work for 12 . however. Since a general functional form is currently unavailable for both coefficients Ck and Cε 1 . This issue should be addressed in future studies. ACKNOWLEDGMENTS Dr. is currently available. which can be called “complete”. to the large-scale turbulence structure. The general conclusion is that a very good agreement between experimental and calculated profiles of the mean velocity.not require any modification. Hussaini (School of Computational Science.
A. 3. v. and Bézard. E. M. Appl. 15(2). Adv. 4 Apsley. S. On ability of standard k-ε model to simulate aerodynamic wakes. 8 Int. P.. 38-54. Poroseva. J. 5 Rogers. pp. 2 Launder. Quart.9(1).. Fluids. 9 turbulent flows. D. W. 1994. B. 1945.) 12 Demuren. v. D. NASA-Ames/Stanford University. Heat and Mass Transfer (Nagoya. Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc. Self-similarity of time-evolving plane Lumley. and Leschziner. K.. P. S. Letters in Heat and Mass Transfer. 487-493. 1997. J. arXiv: physics/0611262. D. Advanced turbulence modelling of separated flow in a diffuser. Modeling the “rapid” part of the velocity/pressure-gradient correlation in inhomogeneous turbulence. v. 2000. The prediction of laminarization with a two-equation model of turbulence. 1998. 1978. I. Y. Fluid Mech. pp. M. v. M. V. R. Rogers. Direct simulation of a self-similar turbulent mixing layer.. Math. 903-923. M. 367. J. 6. and Moser. Heat Mass Transfer. B. 10 Chou. 81-112. (Poroseva.. M. S. D. and Ewing. v. pp. B. 2000. A. pp. S. pp. Japan). M. Mech. 1972. 464-470. The third Poroseva. pp. Center for Turbulence Research. 1996. and Sharma. 1. pp. v. Symposium on Turbulence. O. 7 123-177. On modeling pressure 13 . Rogers. V. 3 Cousteix. Annual Research Brief 2001. H. v. and Launder. Modéles de turbulence: principes et applications.. R. E. 2001. 11 Poroseva. Int. On velocity correlations and the solutions of the equations of turbulent fluctuation. V. J. Computational modeling of turbulent flows. W. and Aupoix.. S. Flow.367-374. M..131-138. Modeling the “rapid” part of the velocity – pressure-gradient correlation in inhomogeneous turbulent flows. 6 Moser. P. L. New approach to modeling the pressure-containing correlations. and Lele. pp. pp. V. 255-288. REFERENCES 1 Jones. 1974.publication. In: Proceed. v. D. Turbulence and Combustion. 2006. Phys. B.1-32. the 16éme Congrés Canadien de Mécanique Appliquée. 2001. Durbin. CFD Journal. 63. Appl. pp. 18.
No F259/a/3/. AIAA Journal. Foundn. CTR Summer Program 1996.. 1998. 26(11). B. Int. S. D. 1995. R. Thesis. and Aupoix. U. Doc.. pp.diffusion in non-homogeneous shear flows. Messing. 13 Wilcox. v. 1997. Development of the k-ϕ turbulence model. France. 14 Cousteix. C. 18 Buice. R. K..18. 16 pp. 14 . Experiments in Fluids. V.. Theoretical and experimental studies on isothermal.. Thermosciences Division. and Driver.464-472. velocity measurements for turbulent separated and near-separated flows over solid waves. Mech. C. and Eaton. University Illinois at Urbana (USA). In: Proceed. and Reynolds. turbulence modeling for wall-bounded flows. 17 Jovic.-B/Fluids.. W.. pp. 21. v.1299-1310. 1995. C. expanding swirling flows with application to swirl burner design. v. Experimental investigation of flow through an asymmetric No. J. P. Stanford University (USA). Department of Mechanical plane diffuser. 599-605. A. Structure-based Durbin. TSD-107. v. Ph. L. Reassessment of the scale – determining equation for advanced turbulence models. pp. Saint--Martin. A. Reynolds number effect on the skin friction in separated flows behind a backward-facing step. Results of the NFA 2-1 Investigation.A. 2000.. Bézard. Kuzan. 33. Grenoble. 1988. 21 Tsinober. S. J. Flame Res. Intl. J. Turbulent Shear Flows. Heat Fluid Flow. C. Langer. D. Bortz. Haire. Department of Chemical Engineering.659-664. Separated flow computations with the k-ε-v2 model. Report 19 Engineering. 20 Hagiwara. 1986. J.D. pp. H. v. S. the 11th Symp. 63-74. 1997. Is concentrated vorticity that important? Euro. 17(4). 15 Kassinos. 421-449. AIAA Journal. D. C. 1986. and Weber. J. pp. A. S.
5 (0.2 1.6) 1.TABLE 1.6 boundary layers β = 19.85 0.1.5 0.9 TABLE 2.6 1. The value of the model coefficients Ck and Cε 1 in free shear flows and equilibrium boundary layers ( β is the pressure gradient parameter).7 (0.2 1 Cε1 Ck 0.6 0. Values of the coefficients Ck and Cε 1 in separated flows shown in Fig.8 1.85 0. not chosen.8 β =0 2. flow Cε1 Ck backstep diffuser wavy channel combustion chamber 1.6 1. flow wake mixing layer plane jet 2.9 0.6) 15 .12 1 round jet 1.5 0. The values in the parenthesis are set.
16 . (d) combustion chamber (black vertical lines show the measurement locations). (c) wavy channel. (b) diffuser.a) b) c) d) Figure 1 Test flows: (a) backstep flow.
< v 2 > − f model (dash-double-dotted lines). (b) x / H = 4 . Notation: experimental data (circles).a) b) c) Figure 2 Backstep flow: velocity (top) and turbulent velocity energy (bottom) profiles. (c) x / H = 6 . the LS model (dashed lines). Locations: (a) x / H = −3 . 17 . the LS-RPD model (solid lines).
Figure 3 Backstep flow: skin friction coefficients. (See notation on Fig. 2.) 18 .
a) b) c) Figure 4 Diffuser: velocity (top) and turbulent velocity energy (bottom) profiles. (c) x / H = 32 . (b) x / H = 28 .) 19 . (See notation on Fig. 2. Locations: (a) x / H = 24 .
Figure 5 Diffuser: skin friction coefficients.25 . (See notation on Fig.) 20 . (b) x / H = 0. 2. Locations: (a) x / H = 0.75 . (See notation on Fig. 2.) a) b) Figure 6 Wavy channel: axial velocity profiles.
(b) r / R = 1.68 .6 . (c) r / R = 3.) 21 .7 . Locations: (a) r / R = 0. 2. (See notation on Fig.a) b) c) Figure 7 Combustion chamber: axial (top) and swirl (bottom) velocity profiles.

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