Numerical study of Jet In Cross-Flow
4
SOME of the recent work on JICF involves numerical simulations. The description of turbulent ows using Computational Fluid Dynamics (CFD) may be achieved using three levels of computations: Reynolds Averaged Navier-Stokes (RANS), Large Eddy Sim- ulation (LES) and Direct Numerical Simulation (DNS). The shortcomings of RANS, where the eects of all turbulence scales are modeled, in JICF have been argued by Chochua et al.
(2000) and Acharya et al. (2001). In contrast to RANS, LES, capturing the dynamics of the large eddies while modeling the eects of the smaller eddies, have provided better agreement with experimental data. Yuan et al. (1999) performed LES of a round Jet In Cross-Flow under comparable conditions to experiments by Sherif and Pletcher (1989).
The LES was shown to reproduce the large scale coherent structures observed experimen- tally. Schlüter and Schönfeld (2000) compared the results of their LES with experimental velocity proles of Andreopoulos and Rodi (1984) and scalar elds of Smith and Mungal (1998), and obtained satisfactory agreement. In addition to all the foregoing, Muppidi and Mahesh (2007) performed DNS at conditions of experiments by Su and Mungal (2004).
The experimentally validated computation of the full instantaneous Navier-Stokes equa- tions was directed towards quantities not available from experiments.
Considering the high computational cost of DNS, the diculty to predict the JICF using RANS computations reported in literature (Chochua et al. (2000) and Acharya et al.
(2001)) and the unsteady ow dynamics brought forward in Chapter 3, a LES approach was decided on. Primarily conned to research CFD codes, in the last years, the ma- jor commercial CFD code vendors have begun oering LES capability in their products.
STAR-CD version 4.06 has been used for this study. As outlined in Paragraph 2.3, the numerical study covers two cases. The chapter is introduced by the experimental ow congurations. In Paragraph 4.2, the computational method is addressed. The governing equations are reviewed, the grid characteristics are detailed and the boundary conditions and simulation parameters are given. The numerical results are presented in Paragraph 4.3.
The paragraph begins with an error analysis and estimation to assess the commercial CFD solver capability and grid quality. Next, the experimental observations in Paragraph 3.1.4 are complemented. Following, the numerical study, matching the experimental conditions of Su and Mungal (2004) (r = 5.7), examines the mediumr JICF ow topology. Besides
63
comparing to the experimental results, the commercial code is also assessed by comparing the Su and Mungal (2004) test case to DNS results by Muppidi and Mahesh (2007). A fundamental deviation from the r = 1ow physics is established. A comparison between ther = 1and r= 5.7 mixing process and eciency ends Paragraph 4.3. The signicance of looking into the JICF ow physics on a scaled up geometry is assessed in Paragraph 4.4.
4.1 Experimental test cases
In Chapter 2, the conclusion for two separate experimental test cases is drawn. That is to say, SPIV on a scaled geometry and measurements by Su and Mungal (2004).
For the scaled geometry test case, reference is made to Paragraph 3.1.2. Figure 4.1a summarizes the experimental set-up.
Su and Mungal (2004) measured the planar scalar mixing in a JICF with r= 5.7and Rej = 5000. They seeded the jet air with acetone and made LIF/PIV measurements of the scalar and velocity eld. The Schmidt number of the system is 1.49. The tunnel cross- ow velocity has a peak value of U∞ = 2.95 m/s and the 80% boundary layer thickness is 1.32d at the location of the centre of the jet exit in absence of the jet. The jet nozzle is a simple pipe with 4.53 mm inner diameter and 320 mm length. No jet exit velocity prole is provided. Nonetheless, in the absence of any cross-ow, fully developed pipe ow conditions are expected at the jet exit. The ow conditions are summed up in Figure 4.1b.
The authors provide detailed experimental scalar proles at a few vertical (y/rd= 0.1, 0.5, 1 and 1.5) and stream-wise stations (x/rd = 0.5, 1, 1.5 and 2.5) on the centre- plane and o-centre measurement planes. Scalar eld results are normalized by the scalar concentration value in the jet nozzle, C0. For each image in the jet centre-plane, z = 0, the jet potential core is in view andC0 is determined directly. For the o-centre planes, Su and Mungal (2004) extrapolated known values from the z= 0 planes. As a consequence, only centre-plane results can be compared.
(a) (b)
Figure 4.1: Experimental test cases: (a) scaled geometry ow conguration (b) Su and Mungal (2004) ow conguration
4.2 Computational method
This paragraph turns to the simulation approach. Details of the computational method are provided. A concise description of STAR-CD's LES implementation is given. The principal focus of the paragraph is on the grid generation and a generic meshing guideline for JICF is proposed. Finally the boundary conditions and simulation parameters are addressed.
4.2. Computational method 65
4.2.1 Governing equations for uid ow
The LES model is based on a spatial lter. A ltered i.e. resolved or large scale variable, denoted by an overbar, is dened as:
f(x, t) = Z
D
f(x0, t)G(x,x0)dx0 (4.1) whereD is the entire computational domain andG is the lter function. The exact form of the lter function is not important within the context of the constant coecient LES model used in STAR-CD as the application of the lter is never computed (CD-Adapco, 2008a). The velocity can then be decomposed into resolved and sub-grid scale components:
ui(x, t) =ui(x, t) +ui,SGS(x, t) (4.2) Applying the ltering operation to the Navier-Stokes equations, this yields the following LES equation:
∂(ρui)
∂t +∂(ρuiuj)
∂xj =−∂p
∂xi +∂τij
∂xj (4.3)
The ltered Navier-Stokes equation, written above, governs the evolution of the large, energy-carrying scales of motion. The eect of the small scales appears through a sub-grid scale (SGS) stress term:
τSGS,ij =ρ(uiuj−uiuj) (4.4)
To model the sub-grid stress tensor, the eddy viscosity hypothesis is used to relate the Reynolds stresses to the resolved rate of strain, Sij:
τSGS,ij =−2µt(Sij −1
3Skkδij) +2
3ρkSGSδij (4.5)
wherekSGS is the sub-grid scale turbulent kinetic energy dened as
ρkSGS =τSGS,kk/2 (4.6)
To close the problem, an implementation of the constant coecient Smagorinsky model is used:
µt=ρCs2∆2 Sij
Sij (4.7)
kSGS= 2CI∆2 Sij
2 (4.8)
where Sij
represents the Frobenius norm of the strain rate tensor, and∆is approximated by the cubic-root of the cell volume. By default, the parameter Cs2 is taken to be the square of the classic Smagorinsky constant Cs = 0.18 andCI is set at 0.202. It was found by Deardo (1970), that in the presence of shear the coecient must be reduced. Values of the order of Cs ≈ 0.1 are reported on. Accordingly, in a rst attempt, Cs was set at 0.14. Details of the Smagorinsky model, as well as more sophisticated LES models, can be found in Benocci et al. (2006). By analogy, sub-grid mixing is modeled by an eddy diusitivity approach with a turbulent diusitivity based on the turbulent viscosity of the sub-grid stress model and a constant Schmidt number:
uiYm−uiYm= νt Sct
∂Ym
∂xi (4.9)
4.2.2 Grid characteristics
Driven by moderate consumption of computational power, an eort was made with respect to the number of cells. The LES by Yuan et al. (1999) and DNS by Muppidi and Mahesh (2007) were used as reference for comparison. Yuan et al. (1999) report on LES at r= 2 and Rej,∞ = 1050. They used 1.3 106 control volumes to discretize the 13.7d×8d×9d computational domain, of which approximately 45% were placed directly downstream of the jet exit. Muppidi and Mahesh (2007) computed the Su and Mungal (2004) ow con- guration on10 106 cells covering a domain of32d×64d×64d.
Dowling (1991) suggested that for round jets the highest dissipation rates occur at the Taylor scale, with aRe−1/2L dependence. He physically interpreted the Taylor scale as the length scale arising from a balance between the local outer-scale strain and diusion. The length scale dependent only on this strain rate and the viscosity is:
λ∝(Lν/U)1/2 =LRe−1/2L (4.10)
He concluded that the highest dissipation rates occur in regions experiencing the strain rate of outer, or energy-containing, ow scales.
Su and Clemens (2003) propose for planar jets aRe−3/4dependence which is consistent with the hypotheses of Kolmogorov and Batchelor, with
λ= ΛLRe−3/4L (4.11)
Unfortunately the scaling coecientΛ appears to be geometry and Re dependent.
Based on this information, the grid is build based on the Taylor micro-scales, with the nal view of facilitating an a priori mesh resolution determination for LES of JICF.
For Homogeneous Isotropic Turbulence (HIT), the turbulence dissipation length scale is approximated by
λν =√
15LRe−1/2L (4.12)
An extension outside HIT assumes a relation between the respective velocity components and length scale directions. That is not obvious for JICF but will be seen to be acceptable as a meshing guideline.
Additionally, to a rst approximation, the strain diusion mechanism should dier between the velocity and scalar gradient elds mainly in the diusivities, which is accounted for by the Schmidt number. The scalar dissipation length scale is expressed by
λD =√
15L(ScReL)−1/2 (4.13)
Found on those considerations and the generic JICF topology reported in Table 3.6, the grids were built as follows. Applicable to low to mediumrJICF, the mesh construction for the scaled geometry is illustrated. On the one hand, to assure good cell quality, a structured mesh was opted for. On the other hand, to ensure the most suitable grid resolution, a multiple block approach was adopted. Figure 4.2 depicts the numerical domain (blue lines) and the cell pattern placed on the jet. x, y and z-coordinates, with reference to the jet exit, are marked in red. Along x and z the numerical domain spans the scaled geometry, 16.66d in the stream-wise direction and 9.25d in the span-wise direction. The upper boundary is positioned at y/rd= 8.5. Consulting the jet penetration in Table 3.6, no jet-wall interference is to be expected. In addition, the computational domain includes a2dlong jet pipe, allowing the ow to develop naturally as the jet emerges into the cross- ow. The cell pattern placed on the jet is composed of a set of four blocks: jet pipe
4.2. Computational method 67
Figure 4.2: Grid topology for the scaled geometry ow conguration
(x/rd∈[−0.5,0.5]), jet exit (x/rd∈[−1,1]∈/ [jet pipe]), near (x/rd∈[1,3]) and far eld (x/rd ∈[3,13.33]). The jet pipe mesh is seen to cut through the free-stream grid. Block dimensions are set in accordance with Table 3.6. For example, the jet penetration and core size, equating four times the vortex core size atx/rd= 3,2.5rdand1.6rd respectively, x the upwind block geometry. As such, the near eld block sizes with2rd×3rd×2rd.
The main block resolution estimates (Eqs. 4.12 and 4.13) are given in Table 4.1. In the stream-wise direction ReL is based on the velocity magnitude |U|and ow widthδ. The vortex core size rc and the up-wash velocityUc provide for the ReL ow variables in the
Table 4.1: Grid stream-wise and radial resolution estimates
x/rd U1 L2 ReL λν x/dν λD x/dD
m/s mm µm µm
Stream-wise
0 4.20 10 2710 743 0.074 743 0.074 1 3.75 10 2420 787 0.079 787 0.079 3 2.50 20 3226 1362 0.136 1362 0.136
Radial
1 0.50 3 97 1180 0.118 1180 0.118
3 0.25 4 65 1927 0.193 1927 0.193
1|U|and Uc respectively
2δ and rc respectively
(a) (b)
(c)
Figure 4.3: Grid topology details for the scaled geometry ow conguration: (a) wall boundary mesh;
(b) jet pipe and exit mesh; (c) near eld mesh
radial direction. As can be seen from Table 4.1, the Taylor micro-scales dictate a nearly isotropic grid. To ensure a smooth connection in between blocks, the resolution for block 1, 2 and 3, in the order given, were set at x/d = 0.0625, 0.0625 and 0.1275. Zoomed in details are provided in Figure 4.3. The jet and cross-ow boundary layer were meshed rst.
Composed of 10 cells and resolving the ow down to the wall(y+ = 2)(Figures 4.3a and b), it is likely that virtually all scales of the uid motion are resolved and that the sub-grid scale contributions to the turbulent stresses are negligible, compensating for the Standard Smagorinsky model performing poorly near the solid walls (Sagaut, 2006). Second, the
Figure 4.4: Grid topology for the Su and Mungal (2004) ow conguration
4.2. Computational method 69
jet was meshed (Figure 4.3b). The most appropriate cell shape in STAR-CD are cuboids (CD-Adapco, 2008b). As such, a O meshing technique was applied. Third, the blocks were meshed and smoothly connected (Figures 4.3b and c). This core grid was allowed to grow linearly outwards at varying ratios. The resulting grid is composed of6 105 cells, of which approximately 70% are placed on the jet. At similar ow conditions and comparable domain characteristics, a reduction in cell number by a factor of 2 with respect to the numerical study by Yuan et al. (1999) is obtained. Although attractive at rst sight, the grid quality is addressed in Paragraph 4.3.1.
Similarly, the experimental set-up of Su and Mungal (2004), spanning a domain of 25d×20d×20d, was meshed (Figure 4.4). The meshing technique only draws apart by an intermediate block connecting the jet exit block (x/d∈[−1,1]∈/ [jet pipe]) to the near eld block (x/rd∈[1,3]). 8.5 105 cells come about for the computational grid. Aware that a comparison to DNS does not hold, the computational domain of Muppidi and Mahesh (2007) gives the impression to be over sized.
4.2.3 Boundary conditions and simulation parameters
In general, the ows in the main section and inow for the pipe are driven by xing the inow velocity proles while matching the experimental conditions. In particular, in absence of the jet velocity prole in the paper by Su and Mungal (2004), a mean turbulent pipe ow is prescribed, based ondand the bulk velocity (Uj,max= 1.5Uj,bulk). In addition, a scalar substitutes for the acetone vapor seeded air jet. On the lateral and top surfaces, free-slip boundary conditions are prescribed, while on the bottom, a no-slip boundary condition is enforced. The outow condition is set to a zero gradient for all ow variables.
A Shear Stress Transport (SST) (Menter, 1994) RANS simulation was run rst and set as initial eld for the LES simulations. Figures 4.5a and b depict theV-velocity eld for the scaled geometry and Su and Mungal (2004) ow conguration respectively. No articial vorticity was introduced into the domain. In Figures 4.5c and d, the instability of the jet and cross-ow mixing layer is seen to initiate the transient. The Su and Mungal (2004) ow congurationv-velocity eld corresponds to a time interval of1.25 ms. Unlike, the beginning of the transient in the scaled geometry ow conguration is observed from 25 ms on. Besides the retarded transition to unsteady ow behavior and the dissimilar v-velocityelds, the relative pressure with respect to atmosphericprelin Figures 4.5e and f is an indication of potentially diering instability mechanisms. On the one hand, the−prel signature in Figure 4.5e seems congruent with the observed ow topology in Chapter 3.
On the other hand, the alternating−prel and +prel traces on the upwind side of the jet in Figure 4.5f, as their prompt initiation, suggest Kelvin-Helmholtz like ow structures.
Time advancement was implicit, central dierencing was used for the convective terms in the momentum equation and the SIMPLE algorithm was used for pressure-velocity coupling. Values of model parameters recommended for LES applications in STAR-CD are provided in CD-Adapco (2008c). The time step was optimized iteratively in order to achieve convergence at every time step with ve outer iterations, leading to a non- dimensional time step in the order of 0.03d/U∞ for both ow congurations, giving a maximum CFL number of 4. The solution is advanced in time to two ow-through times to allow the initial transients to exit the domain before computing time averaged statistics.
Average quantities are measured over a time span of two ow-through times.
(a) (b)
(c) (d)
(e) (f)
Figure 4.5: Large Eddy Simulation analysis preparation: (a) scaled geometry ow conguration RANS initial V-velocity eld; (b) Su and Mungal (2004) ow conguration RANS initial V-velocity eld;
(c) scaled geometryv-velocityevolution,t= 25 ms; (d) Su and Mungal (2004) ow congurationv-velocity evolution,t= 1.25 ms; (e) scaled geometryprel evolution,t= 25 ms; (f) Su and Mungal (2004) ow con- gurationprelevolution,t= 1.25 ms
4.3 Results
Prior to give thought to the ow physics, the CFD results will be evaluated. The various types of error in the numerical solution of uid ow problems and the common practice for CFD uncertainty analysis are gone over. Referring to Ferziger and Peric (2002): Any
4.3. Results 71
reasonable estimate of numerical errors is better than none; and numerical solutions are always approximate solutions, so one has to question their accuracy all the time, the general rules for validation are revisited, made specic and applied to the obtained results.
4.3.1 Error analysis and estimation
In addition to the errors that might be introduced in setting up the boundary conditions, numerical solutions always include three kinds of systematic errors (Ferziger and Peric, 2002):
Modeling errors, which are dened as the dierence between the actual ow and the exact solution of the mathematical model;
Discretization errors, dened as the dierence between the exact solution of the conservation equations and the exact solution of the algebraic system of equations obtained by discretizing these equations, and
Iteration errors, dened as the dierence between the iterative and exact solutions of the algebraic equations systems.
Error analysis should be done in an order reversed from the order in which they were introduced above. Assuming that the established STAR-CD code uses appropriate con- vergence criteria by default, the validation of the CFD results includes the analysis of discretization and modeling errors. On the one hand, discretization errors can only be estimated through a grid-dependence study. On the other hand, to estimate the modeling errors, reference data are needed to compare them with the numerical solutions.
Experimental and/or accurate simulation data answer the need for quantifying the modeling errors. However, the computational resources in the early stages of the research, did not allow to perform mesh renement studies. As such, on paper, the total simulation error arising from the modeling and discretization can be estimated only. Although satis- fying the motivation of this qualitative analysis and the response of the system to future design changes, apart from the comparison to experimental data, the numerical errors are evaluated through the grid adequacy and the resolution after having carried out the LES.
With respect to the grid quality, the grid lines at cell corners are mostly orthogonal and the angle between the line connecting neighboring cell centers and the cell-face normal is for the most part at 90°, making certain of an optimized grid. To assess the resolution a posteriori, Davidson (2010) identies various quantities: energy spectra, dissipation energy spectra, two-point correlations, the ratio of SGS shear stress to resolved shear stress, the ratio of the SGS viscosity to the molecular viscosity and the ratio of the SGS dissipa- tion due to the resolved uctuating velocity gradients to that due to the mean velocity gradients. Energy spectra and the fraction of the kinetic energy within the resolved ow are chosen and applied in the subsequent paragraphs. For the scaled geometry test case, rst, the ratio of resolved toresolved + SGSkinetic energy and extended spectral densities of uctuating velocities are presented. Second, the numerical and experimental data are confronted. Opposite, for the Su and Mungal (2004) ow case, the paragraph is introduced by equating the computations and measurements. Afterwards, dierences are evaluated by the corresponding grid resolution computations.
Scaled geometry ow conguration
The grid quality is rst assessed by the ratio of resolved to resolved + SGS kinetic en- ergy at time step 14000, in other words at t = 1.4 s. Afterwards, the temporal signal is monitored at three locations and the Fourier transform is computed. The monitored cells are placed at locations of high turbulence and extend over block 2 and 3. Figure 4.6 overlays the turbulent kinetic energy by 90% ratio isolines. In block 3, sensor 1 and 2 are located at x/rd = 3.75, y/rd = 1.8125,z/rd = 0.3125 and x/rd = 3.75, y/rd = 0.9, z/rd= 0.1875respectively. Sensor 3, part of block 2, was placed atx/rd= 1.5,y/rd= 0.9, z/rd= 0.3125.
In Figure 4.6 the ratio of the resolved turbulent kinetic energy to the total turbulent kinetic energy is seen to be greater than 90% in regions with high turbulence. Using the guideline by Pope (2000), suggesting a ratio greater than 80%, the results from this LES run hint a well-resolved ow.
To strengthen the assumption, the one-dimensional energy spectrum of the velocity uctuations along thex, y, z-axisat the three locations are given in Figure 4.7. All spectra exhibit a decay which is quite similar to the Kolmogorov's−5/3 decay rate prediction in the inertial range of turbulence, giving further evidence that the grid resolution is ne enough to resolve a portion of the inertial range. Unrelated to the error analysis and estimation, it is worthwhile noticing that the energy in the three directions is in the same order of magnitude at monitoring points 2 and 3, which indicates that the turbulence there is three-dimensional. Contrary to locations 2 and 3, at point 1, the energy in thez-direction is orders of magnitude smaller than the other two components. The ow expresses itself mainly in the spectra foru andv, remaining more or less two-dimensional.
Judging the resolution sucient, the validity of the model is assessed in Figures 4.8, 4.9 and 4.10. First, the jet trajectory is looked at. Second, consideration is given to the raw velocity magnitude and the magnitude of the velocity with the cross-ow speed subtracted.
In the end, the turbulent resolved normal stress components are compared. Throughout this dissertation, the experimental results are either colored in red and the numerics are represented in blue or numerical results are overlaid in lines on experimental color maps.
The validation is entirely established on centre-plane data and only a part of the LES domain is shown in order to make a one to one comparison with the PIV data domain.
Figure 4.6: Scaled geometry ow conguration grid quality
4.3. Results 73
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
Figure 4.7: Energy spectra versus frequency: (a) Eu0u0(f) at location 1; (b) Ev0v0(f) at location 1;
(c)Ew0w0(f)at location 1; (d)Eu0u0(f) at location 2; (e)Ev0v0(f)at location 2; (f)Ew0w0(f)at location 2; (g)Eu0u0(f)at location 3; (h)Ev0v0(f)at location 3; (i)Ew0w0(f)at location 3
As a starting point, the quality of comparing the PIV and LES data for the purpose of model validation needs attention. It is agreed that mean ow quantities match. Quantities like Reynolds stresses and turbulent kinetic energy are harder to compare. Constructed from ow uctuations, they are sensitive to resolution dierences between measurements and simulations. With the same resolution data sets, the PIV analysis contains an in- terpolation procedure typical of the use of interrogation windows, whereas the LES data are resolved on the grid. Talstra et al. (2006) report LES data exceeding the PIV data values up to a factor 2. For a fair comparison, an adaptation should be made to the LES data. Despite these considerations, no adjustement lter was applied to the LES velocity uctuations. As a matter of fact, the PIV and LES data are expected to dier in their quantitative characteristics, bringing the resemblance down to a qualitative nature.
Figure 4.8: Comparison of centre-streamline trajectory: LES (-); Experiments (e)
(a) (b)
Figure 4.9: Comparison of mean velocity quantities: (a)|U|; (b)|U−U∞ex|: LES (lines); Experiments (map)
The centre-streamline trajectories are plotted in Figure 4.8. A nice t is observed. To visualize the trajectory of the jet, the centre-streamlines are included in all subsequent plots. In Figure 4.9 |U| and |U −U∞ex| are overlaid, to assess the jet like nature in the near eld and the wake like behavior in the far eld respectively. In the near eld (Figure 4.9a),|U|is in good agreement. In the wake portion, as evidenced by Figure 4.9b,
|U −U∞ex|decays to fast. Therefore the LES appears to be too dissipative. To compen- sate for the excessive damping, a reduction of Cs in the far eld should be imposed. The centre-plane mean turbulent stress componentshu0u0i,hv0v0iandhw0w0i are represented in Figure 4.10. While, as anticipated, the lower PIV resolution in blocks 1 and 2 as well as the similar resolution in block 3 results in an overestimation of the normal stress components,
(a) (b)
4.3. Results 75
(c)
Figure 4.10: Comparison of turbulent normal stress components: (a)hu0u0i; (b)hv0v0i; (c)hw0w0i: LES (lines); Experiments (map)
the regions of turbulent kinetic energy production are well reproduced nevertheless. In addition, the structure is consistent with the experimental observations. So, the turbulent kinetic energy at the jet centre-line is primarily composed byhu0u0i(Figure 4.10a), suggest- ing a penetration of the cross-ow into the jet. hv0v0i and hw0w0i are located downstream of the edge of the jet, covering the entire lower part. These large co-located regions of hv0v0i andhw0w0i (Figures 4.10b and c) have the signature of a three-dimensional waving.
Per se, although lacking quantitative compatibility, the qualitative coherence indicates the correct reproduction of the ow eld.
Su and Mungal ow conguration
As opposed to the scaled geometry ow conguration, the experimental and numerical re- sults are correlated rst. Afterwards discrepancies are related to the grid or uncertainties in the experimental set-up. Although the main driver for the Su and Mungal (2004) test case is the substantiation of a reliable scalar eld, the model validation is preceded, analo- gous to the previous paragraph, by mean and uctuating velocity quantities. DNS data by Muppidi and Mahesh (2007) is overlaid, whenever available. Both computational methods
Table 4.2: LES and DNS equivalence
Argument LES DNS
Free-stream X X
Jet Scalarair, ρexp Air1
Mean turbulent velocity prole Separate simulation of a fully turbulent pipe ow
Domain 25d×20d×20d 32d×64d×64d
2dpipe 2dpipe
Mesh Structured, cubical elements Unstructured, hexahedral ele- ments
8.5 105 cells 10 106 cells
Numerical scheme STAR-CD Mahesh et al. (2004)
Standard Smagorinsky
1Velocity proles are normalized to account for the dierence in densities.
Figure 4.11: Comparison of centre-streamline trajectory: LES (-); DNS (5); Experiments (e)
are summarized in Table 4.2. Slightly diering boundary conditions and computational domains were applied and as such a perfect equivalence of the computational results is not guaranteed. For the experimental set-up, the reader is referred to Figure 4.1b. A check mark indicates a perfect match.
As mentioned earlier, in the absence of the normalization rule for the o-centre-plane scalar elds, only centre-plane results are compared. Stream-wise, x/rd = 1, 1.5 and 2.5 positions are looked at. Along the y-axis, y/rd locations ranging from 0.1 to 1.5 are considered. The symbology is analogous to the previous paragraph. Experimental results are colored in red and, in general, the numerical outputs are depicted in blue. In particular, the LES results are marked by lines and the DNS results by open circles.
The centre-streamline trajectories are depicted in Figure 4.11. The trajectory slightly underestimates the experimental results. In the DNS computations, the under-shoot is seen to be even more pronounced. The mean velocity variables|U|and|U −U∞ex|are shown in
(a) (b)
Figure 4.12: Comparison of mean velocity quantities: (a) |U|; (b) |U−U∞ex|: LES (-);
Experiments (e)
4.3. Results 77
(a) (b)
Figure 4.13: Comparison of turbulent normal stress components: (a) hu0u0i; (b) hv0v0i: LES (-);
DNS (5); Experiments (e)
Figure 4.12. As in the scaled geometry ow case (Figure 4.9), on the one hand, a nice match is observed for|U|(Figure 4.12a). On the other hand,|U−U∞ex|(Figure 4.12b) becomes inferior, emphasizing the motivation for an adaptiveCs. The mean turbulent normal stress components hu0u0i and hv0v0i are represented in Figure 4.13. Fluctuations in the vertical direction at the jet exit (Figure 4.13b) enforce the feeling that the turbulence over the jet is generated by Kelvin-Helmholtz like span-wise vortical structures. This concept, although not yet veried, then begins to produce stream-wise uctuations (Figure 4.13a). At the y/rd = 0.5 location, with the exception of the lower peak in hu0u0i downwind of the jet exit, all stress components are well reproduced by the simulations. To investigate those apparently smaller scaled trailing edge ow structures, the one-dimensional energy spectra of the velocity uctuations along the x and y-axis at the out of line location are given in Figure 4.14. Unlike the inertial range characteristic decay rate of the Ev0v0(f) power spectrum in Figure 4.14b, in Figure 4.14a no cup-like shape is observed in the Eu0u0(f) power spectrum. Rather than decreasing at a rate of −5/3, the spectrum drops o in a steeper manner from f ≈ 3000 Hz on. Presumably, the grid is too coarse to resolve the large structures. As a consequence, the lower peak in hu0u0i at y/rd = 0.5 is regarded likely. At y/rd = 1, the location of the maximum turbulence intensity where the leading
(a) (b)
Figure 4.14: Energy spectra versus frequency at location x/rd = 0.07, y/rd = 0.5 and z/rd = 0:
(a)Eu0u0(f); (b)Ev0v0(f)
and trailing edge vortices collide, hu0u0i is well captured by the simulation. The higher hv0v0i at the y/rd = 1 location is less conclusive. From one point of view, the deviation could arise from diering resolutions between measurements and simulations. Weighing the single peak divergence and the agreement between the LES and DNS makes this possibility improbable. From another point of view, New et al. (2006) studied the inuence of the jet velocity prole on the characteristics of a round JICF. As a result, they concluded that there is an increase in jet penetration and a reduction in the near eld entrainment of cross-ow with a parabolic JICF in comparison with the corresponding top hat JICF, where the shear layers are thinner. A jet velocity prole with thicker shear layers is less susceptible to instabilities, the shear layer will roll up in less energetic ow structures and the penetration will be higher. Those ndings suggest that the pipe ow in the experiment is not fully turbulent and draws closer to a parabolic prole.
The scalar concentration eld is addressed next. The resemblance of the scalar trajec- tory is examined rst. The experimental and numerical scalar trajectory, dened as the loci of maximum scalar concentration, are drawn in Figure 4.15. The computations are seen to reproduce an almost exact copy of the experimental results. Figure 4.16 compares the mean and uctuating scalar concentration in the near eld. For the mean scalar eld in Figure 4.16a, a very good agreement is observed. In Figure 4.16b, the agreement for the uctuating scalar eld appears reasonable at the rst and third location, while there is some deviation between the simulation and the experiments aty/rd= 1. At the jet exit, mixedness
M I = 1− hc0c0i
C(1−C) (4.14)
is well reproduced, particularly seen in the context of the sharp gradients. At they/rd= 1 location, while the location of the peaks appear to coincide, the peak magnitudes dier.
Mixedness is under predicted at the windward side and over predicted at the downwind side. Unfortunately, no DNS data is available. The discrepancy on the windward side is in accordance with the computed stress components. The larger ow structures, identied by the dierence in hv0v0i, generate a higher entrainment and a lower mixedness. The mean and uctuating scalar concentration in the far eld are compared in Figure 4.17.
As for the near eld, the mean scalar eld (Figure 4.17a) is in very good agreement. For the uctuating scalar eld, two distinct branches are observed in Figure 4.17b. The upper branch reproduces well the downstream continuation of the mixing produced by the leading edge vortices. On the lower branch, the same trend as in they/rd= 1 near eld shows up.
Mixing produced by shear between the jet and the cross-ow and the intermittent CRVP
Figure 4.15: Comparison of scalar centre-line: LES (-); Experiments (e)
4.3. Results 79
(a) (b)
Figure 4.16: Comparison of mean and uctuating scalar concentration in the near eld: (a) mean scalar eld; (b) averaged scalar variance: LES (-); DNS (5); Experiments (e)
is over predicted. In the o-centre proles, Su and Mungal (2004) observe similar trends to those seen in the z = 0 plane, except that the wake-side variance peak is larger. At
(a)
(b)
Figure 4.17: Comparison of mean and uctuating scalar concentration in the far eld: (a) mean scalar eld; (b) averaged scalar variance: LES (-); Experiments (e)
largez/rdvalues, the wake-side variance peak becomes even larger than the windward side peak. Based on this trend, a slight o-centre measurement or slight o-axis experimental set up would explain the dierence in averaged scalar variation on the wake side.
Overall, the agreement between simulations and experiments is quite good for both ow cases. Most discrepancies were identied and related either to the computational method or uncertainties in the experimental set-up. An attempt could be made to adapt the model to the ow by adjusting the numerical constant Cs of the Standard Smagorinsky model.
Constrained by a constantCsthroughout the numerical domain, and with the primary goal of being an optimization on geometry and ow conditions, no case sensitive adaptation of Cs was decided on. The overall agreement in the mean and uctuating velocity and scalar eld, assessed by comparison to experiments, exemplies the ability of the commercial code STAR-CD, with the Standard Smagorinsky model, to replicate adequately the JICF ow eld. After successful assessment of the ability of LES to reproduce accurately the JICF ow physics, in the subsequent paragraph, the dierent vortical structures and their interaction will be looked into in more detail. So far, the preliminary analysis of Figures 4.5, 4.10 and 4.13 indicates diering kinematics for low and mediumr JICF.
4.3.2 Flow physics
The starting point of this paragraph tackles the notion of diering kinematics for both ow congurations as suggested previously. This hypothesis, supported by a side by side mean eld study, leads subsequently to an independent analysis for the scaled and Su and Mungal (2004) ow conguration. The former computations seek to answer the unsettled issues of the experimental investigation. The main focus is on the mechanism for the dynamical generation and evolution of the CRVP. The latter numerical study is mainly directed towards an analysis of the vortical structures.
(a) (b)
Figure 4.18: Comparison of scaled geometry and Su and Mungal (2004) ow conguration topology, y/rd= 0.3,W-velocity,U-velocity isolines (-), V-velocityisolines (-) and pressure isosurface: (a) scaled geometry ow congurationr= 1; (b) Su and Mungal (2004) ow congurationr= 5.7
4.3. Results 81
In Paragraph 3.1.4, vorticity and the second invariant of the mean velocity gradient tensor were consulted to illustrate the vortical structures. In the numerical study, those criteria are combined with low-pressure cores. The addition of pressure isosurfaces also highly aid the visualization and comprehension. Moreover, the three-dimensionality of the computations overcomes the SPIV limit to in-plane velocity components and gradients and allows a full use of theQ-criterion.
Figure 4.18 shows lled contours of mean W-velocity, with mean V-velocity black isolines and red meanU-velocityisolines in overlay, close to the jet exit aty/rd= 0.3. For clarity, thexz-planeis tilted under an angle of 45°. The view is completed by mean pressure isosurfaces. On the one hand, the cross-ow converging upon the span-wise centre-line and a reversed ow region are identied for both ow congurations. On the other hand, the vortex core locations dier fundamentally. Forr = 1, the three distinct ow structures, in- hole vorticity (A), DSSN (B) and hanging vortex (C), observed in the experimental study, are clearly distinguished in Figure 4.18a. For r = 5.7, in Figure 4.18b, one sees a vortex sheet extending from the windward side to the lateral edges of the jet. The vortex sheet is likely to result from the formation of span-wise rollers, in accordance with the hv0v0i distribution in Figure 4.13b. No vortex sheet is observed on the downwind side, indicating a more regular and earlier formation of vortices in the upstream shear layer in comparison with the downstream layer. The vortex sheet signature is in favor of Kelvin-Helmholtz like instability. Intrinsically, Figure 4.18 gives doubtless evidence of diering ow mechanisms.
Details for each conguration are given in the following two paragraphs.
Scaled geometry ow conguration
The object of this paragraph is to extend the conclusions obtained by the experiments developed in Paragraph 3.1.4. Commonalities, in other words, jet oscillation and RLV, will only be looked at briey. The proposed kinematic model will be supported and sup- plemented by the temporal evolution of the vortical structures. On Figure 4.19, the centre- plane is ood by v-velocity and the xz-plane at y/rd = 1.25 is mapped with w-velocity contours. Alternating downstream pairs of v > 0, w < 0 and v < 0, w > 0 (a-a', b-b' and c-c') visualize the three-dimensional waving. The apparently periodic oscillation is
Figure 4.19: Scaled geometry ow topology: instantaneous w-velocity, y/rd = 1.25, instantaneous v-velocity, centre-plane, pressure isosurface and streamlines
Figure 4.20: Low r vortex model: instantaneous z-vorticity, centre-plane, pressure isosurface, +Q-isosurface(blue) and−Q-isosurface(red)
optically enhanced by the two streamlines. The pressure isosurface outlines ring like ow patterns. The z-vorticity and Q-distribution in Figure 4.20 gives a complete picture of a frozen ow topology. The yz-plane is located at x = 0. Positive values of Q are colored in blue and regions of negative Qare represented in red. A vorticity sheet is observed to extend from the vicinity of the jet exit to about 1rd downstream. Beyond this location +z-vorticityjet like and−z-vorticitywake like structures alternate along the jet trajectory.
The nature of the z-vorticity structures is assessed through the Q-criterion and pressure isosurface. On the one hand, the jet like structures coincide with−Q, identifying them as regions of local shear, induced by the cross-ow and jet interaction. On the other hand, the+Qand pressure isosurface perfectly match and identify the rotational stream-tubes, closing on themselves. One note also the experimentally observed S-like ow trajectory in the RLV. The experimental and numerical consistency decidedly allows to reject the most commonly accepted Kelvin-Helmholtz instability ow models (Fric and Roshko (1994), Lim et al. (2001) and Özcan et al. (2005)) and conrm the observations by Camussi et al.
(2002) for low r JICF. Next, the CRVP formation mechanism is investigated using the temporally evolving ow eld.
As a starting point,prel has been tracked simultaneously in four locations (Figure 4.21) over a time span of 0.095 s. At t1, the monitored points are determined by low pressure
Figure 4.21: Spatial locations of relative pressure spectra
4.3. Results 83
Figure 4.22: Temporal evolution of relative pressure
regions. Point A is assigned to the in hole vorticity. Point C is established in the hanging vortex. Point E observes the evolving RLV and point F reects the lateral waving. The evolution of prel in time is drawn in Figure 4.22. The signal coherence in phase links up the in hole vorticity and hanging vortex to the evolving periodical structures cutting through points E and F. Figure 4.23 plots the Fourier transform of the prel signals. A strong periodicity (Figures 4.23a, b and d), except weakened for the in hole vorticity (Figure 4.23c), is observed. The vibrating frequency f of the jet equals 83 Hz, which
(a) (b)
(c) (d)
Figure 4.23: Spectra of relative pressure: (a) RLV (E); (b) Hanging vortex (C); (c) In hole vorticity (A);
(d) Waving (F)
(a) (b)
(c) (d)
(e) (f)
Figure 4.24: Lowrkinematic vortex model: side view,z/rd= 0.2; top viewy/rd= 0.1:t1(a)v-velocity;
(c) concentration; (e) pressure gradient alongx-axis;t2 (b) v-velocity; (d) pressure gradient alongy-axis; (f) concentration, pressure isosurface and streamlines
4.3. Results 85
presented in the non-dimensional form of the Strouhal number, dened by St= f d
U∞ (4.15)
is approximately 0.33. In addition, on the one hand, the far greater signal power of the hanging vortex in Figure 4.23b compared to the spectral power density of the in hole vorticity in Figure 4.23c indicates that the pressure uctuations are greater in the hanging vortex. On the other hand, the spectra of the hanging vortex (Figure 4.23b) and RLV (Figure 4.23a) have similar magnitudes, indicating that the hanging vortex contributes for the largest part in the RLV formation. The dynamic generation and evolution of the vortical structures is analyzed in Figure 4.24. The evolution of the jet is represented by plotting the eld variables at t1 and t2 (Figure 4.22), corresponding to a subsequent low and high pressure signature in the hanging vortex. At t1 (Figures 4.24a, c and e), a v-velocityside view atz/rd= 0.2, scalar concentration andx-pressuregradient top view at y/rd= 0.1, completed by pressure isosurfaces and streamlines are shown in the left column.
In the right column, analogous data, though diering by a y-pressure gradient side view at z/rd= 0.2, is reproduced. Regions of low concentration (w <0.1)are blanked out for better readability. Att1, two RLVs G and H are identied. RLV H branched and evolves downstream. The starting RLV G is generated by the in hole vorticity combining with the hanging vortex. Initiated by the in hole vorticity and feeding on jet uid (streamline 1), as evidenced by the kidney-like scalar eld in Figure 4.24c, the starting vortex legs tilt, combine with free-stream uid owing through the hanging vortex (streamline 2), then bend nearly at a right angle and combine, as manifested by the pressure isosurfaces. The free-stream impacting on the jet, as shown by the depression in thev-velocity prole at the jet exit in Figure 4.24a, is responsible for the tilting and bending. At rst aligning with the free-stream, the vortex legs roll-up around the counter-clock rotating ow structure (Figure 4.24a), triggered by the free-stream and jet interaction. The x-pressure gradient (Figure 4.24e), set o by the low pressure region immediately downstream of the jet exit, forces the jet and vortical structures upstream. At time t2, the in hole vorticity and hanging vortex stand most windward, as indicated by the pressure isosurfaces location in Figures 4.24a and b. Moving upstream, the hanging vortex progressively blocks the jet exit. As a consequence, the jet accelerates and jet uid is drawn in the hanging vortex and towards the vortex legs. The spoon-likev-velocityprole and spreading into the tilted vortex legs in Figure 4.24b illustrate the ow mechanics. The altered v-velocity outline incites a strong y-pressure gradient (Figure 4.24d) on the downstream side. Consequently the vortex legs have the tendency to break down (streamline 3). In addition, the asymmetry J in the scalar eld in Figure 4.24f compared to the regular pattern in Figure 4.24c is a sign of vortex shedding. The faint periodicity of the pressure signature at the in hole vorticity location in Figure 4.22 indicates possible irregularities in the vortex breakdown and shedding process. Finally, the RLV G branches and the formation of RLVs repeats itself.
Su and Mungal ow conguration
Mostly concerned by the Micromix representative r of approximately 1, the analysis of the Su and Mungal (2004) ow conguration is therefore kept descriptive in essence.
Figure 4.25 gives a representation of the hypothesized vortex tube model for r = 5.7.
All vortex tubes (red, black and blue arrows numbered 1, 2 and 3 respectively) of the pressure isosurface are believed to be vortices. Consistency is obtained by z-vorticity in
Figure 4.25: Instantaneousz-vorticity, centre-plane, pressure isosurface and+Q-isocontours
the centre-plane overlaid by +Q-isocontours. Jet shear layer vortices, positively iden- tied by the collocation of high vorticity and +Q-isocontours, appear on the upstream and leeward side. The Upstream Shear Layer Vortices (ULV) (1) wrap around the jet, whereas the Leeward Shear Layer Vortices (LLV) (2) remain in the jet core. The ULV and LLV side arms are tilted and roughly realign with the jet. To further analyze the highly three-dimensional ow eld, instantaneousy-and z-vorticity and scalar concentra- tion superposed by+Q-isocontoursat the AA' intersectiony/rd= 0.55in Figure 4.25 are reproduced in Figure 4.26. The in-plane signature of the vortex tubes of Figure 4.25 are identied by the corresponding numbers. Close to the centre-plane, the side arms of the ULV and LLV are seen in Figure 4.26a to be in the z-direction and to contribute mainly to z-vorticity. Bending towards the jet results in y-vorticity (Figure 4.26b). The ULV are believed to initiate the CRVP, as y-vorticity of the ULV and CRVP are of the same rotation sign. The LLV, carrying y-vorticity of opposite sign, is the origin of a secondary CRVP with rotation sense opposite to that of the primary CRVP. Both CRVP coexist, as
(a) (b)
