James D. Holdeman

Glenn Research Center, Cleveland, Ohio

James R. Clisset

University of Florida, Gainesville, Florida

Jeffrey P. Moder

Glenn Research Center, Cleveland, Ohio

Spreadsheet Calculation of Jets in Crossfl ow:

Opposed Rows of Slots Slanted at 45

NASA/TM—2011-215980

December 2011
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James D. Holdeman

Glenn Research Center, Cleveland, Ohio

James R. Clisset

University of Florida, Gainesville, Florida

Jeffrey P. Moder

Glenn Research Center, Cleveland, Ohio

Spreadsheet Calculation of Jets in Crossfl ow:

Opposed Rows of Slots Slanted at 45

NASA/TM—2011-215980

December 2011

National Aeronautics and

Space Administration

Glenn Research Center

Cleveland, Ohio 44135
Acknowledgments

The authors would like to thank Mr. Richard E. Walker (Aerojet Liquid Rocket Company, retired), Dr. Ram Srinivasan (then 

of Garrett Turbine Engine Company), Professor William E. Lear of the University of Florida, Mr. Timothy D. Smith, and

Dr. C. John Marek (retired) of the NASA Glenn Research Center for their contributions to the NASA jets in crossfl ow (JIC) 

empirical model. The authors also wish to extend thanks to Messrs. David S. Liscinsky of the United Technologies Research 

Center, Robert Reynolds of Honeywell Engines, Richard E. Walker of Aerojet Liquid Rocket Company (retired), and Professors

Dilip R. Ballal of the University of Dayton, John F. Foss of Michigan State University, William E. Lear of the University of 

Florida, and Hukam C. Mongia of Purdue University for reviewing these results.

Available from

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National Technical Information Service

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Available electronically at http://www.sti.nasa.gov

Trade names and trademarks are used in this report for identifi cation 

only. Their usage does not constitute an offi cial endorsement, 

either expressed or implied, by the National Aeronautics and 

Space Administration.

Level of Review: This material has been technically reviewed by technical management. 

Supplementary Notes

Jim Holdeman, retired. A supplemental Microsoft Excel (Microsoft Corporation) spreadsheet TM-2011-215980-SUPPL1.xls is 

available from the NASA Center for AeroSpace Information, 443-757-5802.
NASA/TM—2011-215980 1 

Spreadsheet Calculation of Jets in Crossflow: 

Opposed Rows of Slots Slanted at 45

James D. Holdeman1

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

James R. Clisset 

University of Florida 

Gainesville, Florida 32611 

Jeffrey P. Moder 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

Abstract 

The purpose of this study was to extend the NASA jet-in-crossflow empirical model, implemented in 

the Excel (Microsoft Corporation)spreadsheet documented in NASA/TM—2005-213137, to the case of 

jets entering the mainstream flow from opposed rows of 45 slanted slots. The results in this report were 

obtained using a spreadsheet modified from the one posted with NASA/TM—2010-216100. The 

empirical model in this spreadsheet was not changed from the one posted with NASA/TM—2005-

213137. The modifications in the current spreadsheet, which affect only 45 slanted slots and without 

which distributions for many of the cases reported here could not be calculated, were: 1) the function 

that causes lateral translation for flow from 45 slanted slots was corrected so distributions for round 

holes (as L/W = 1 slanted slots) do not translate and 2) the capability was added to allow 45 slots to 

slant in either direction so perpendicular configurations, where the slots are angled in opposite directions 

on top and bottom walls, could be calculated. The primary conclusion in this report is that the best 

mixing configuration for opposed rows of 45 slanted slots at any downstream distance is a parallel 

staggered configuration where the slots are angled in the same direction on top and bottom walls and one 

side is shifted by half the orifice spacing. Although distributions from perpendicular slanted slots are 

similar to those from parallel staggered configurations at some downstream locations, results for 

perpendicular slots are highly dependent on downstream distance and are no better than parallel 

staggered slots at locations where they are similar and are worse than parallel ones at other distances. 

Nomenclature 

AJ / AM jet-to-mainstream area ratio = ((/4)W2+(L/W-1)W2) / ((S)(H)) 

C (S/H) (sqrt (J)) 

Cd orifice discharge coefficient = (effective area)/(physical area) 

d actual physical diameter of a round hole 

DR jet-to-mainstream density ratio, J/M

H duct height 

H/ d ratio of duct height to orifice diameter 

 

1

 James D. Holdeman, retired. 
NASA/TM—2011-215980 2 

J jet-to-mainstream momentum-flux ratio, (JVJ

2

) / (UM

2

)

L larger dimension of slot 

L/W ratio of larger to smaller dimension of slot 

S lateral spacing between equivalent locations of adjacent orifices 

S/H ratio of orifice spacing to duct height 

UM unmixed mainstream velocity 

VJ jet exit velocity 

W smaller dimension of slot = d / (sqrt(1+(L/W-1) / (4/))) 

x downstream coordinate; x = 0 at center of the first row of orifices 

y cross-stream coordinate 

z lateral coordinate 

 dimensionless scalar

Introduction 

The extensive studies of rows of non-reacting jets in crossflow (JIC) in rectangular, annular, reverseflow, and cylindrical configurations that are summarized in Reference 1 were motivated by mixing of 

dilution jets in conventional gas turbine combustors. The studies summarized in References 2 to 4 

focused on optimizing the mixing section in the Rich burn/Quick mix/Lean burn (RQL) combustor 

scheme in rectangular, annular, and cylindrical ducts for both reacting and non-reacting flows. 

Later, the NASA JIC empirical model from Reference 1 (without the curvature effects) was 

implemented in the Excel spreadsheet reported in Reference 5. No changes were made to the empirical 

model in the spreadsheets posted with References 6 and 7, and these versions (and the current one) are 

backwardly compatible to the spreadsheet posted with Reference 5. Details discussed in the following 

reports apply to the empirical model used in this report but are not repeated: 1) description of the 

empirical model for a conserved scalar in JIC flowfields (Refs. 1, 5 to 7), 2) spreadsheet specifics (Refs. 

5 to 7), 3) a listing of the correlation equations (Refs. 1 and 5), 4) listings of the original BASIC code 

(Ref. 5), 5) the “closest” experimental data (Ref. 5), 6) contour plots from exported data (Ref. 6), 7) 

comparison of results for aligned jets using both the symmetry and superposition models (Ref. 6), 8) a 

slideshow using profile plots (Ref. 6), 9) the addition of low resolution contour plots within the 

spreadsheet (Ref. 7), and 10) empirical model calculations for several cases of jet mixing in a confined 

crossflow (Refs. 5 to 7). 

The objective of this study was to investigate flow from opposed rows of jets from slots slanted at 

45 from the direction of the mainstream flow. The spreadsheet used in this report is derived from the 

one posted with Reference 7; however, two important changes were made that only affect slanted slot 

results: 1) the function that causes lateral translation for 45 slanted slots was corrected so distributions 

for round holes (as L/W = 1 slots) do not translate and 2) the capability was added to allow 45 slots to 

slant in either direction. Without these modifications, distributions for many of the cases reported herein 

could not be calculated. 

Although experimental, CFD, and empirical model results for single-side injection from slanted slots 

have been shown previously (e.g., Ref. 1), empirical model results for opposed rows of slanted slots have 

not. CFD calculations for opposed rows of slanted slots in a rectangular duct are shown in References 1, 

8, and 9. Experimental results were published in Reference 10 for several configurations of opposed 

rows of 45 slanted slots. 

This report presents results using the NASA JIC empirical model for several configurations of 

opposed rows of 45 slanted slots. 
NASA/TM—2011-215980 3 

Flow Field Model 

Figure 1 is a schematic of the basic flow field for a row of jets in a confined crossflow. For all 

calculations, the dimensionless flow and geometric variables that must be specified are the jet-tomainstream density ratio DR (=J /M), jet-to-mainstream momentum-flux ratio J (=(JVJ

2

) / (UM

2

)),

jet discharge coefficient Cd, orifice-spacing-to-duct-height ratio S/H, duct-height-to-orifice-diameter 

ratio H/d, and the dimensionless downstream distance x/H. For 45 slanted slots one also needs to specify 

the slot aspect ratio L/W and the slant direction. 

Correlation equations were developed for each of the dimensionless parameters needed to define the 

conserved scalar  in all three spatial dimensions. Three-dimensional oblique plots of vertical scalar 

profiles and low resolution contour plots of  are displayed in the spreadsheet. The dependent variable 

is shown on the horizontal axis in the profile plots and  is the plotted variable in the contour plots. The 

vertical and oblique axes in the profile plots are the y and z directions. These are the axes of ordinates 

and abscissas in the contour plots. In the current report, unmixed jet fluid is  = 1 and unmixed 

mainstream fluid is = 0. 

Two effects were noted in the experimental results for one side injection from 45 slanted slots in 

References 1 and 9. These were: (1) the distributions for slanted slots shifted laterally with downstream 

distance and (2) the axes of the kidney-shaped contours was inclined with respect to the direction of the 

injection. The NASA JIC empirical model distributions for 45 slanted slots show the shift, but the 

inclination and asymmetry of the  distribution is not modeled. It was reported in Reference 1 that 

slanted slots seem to result in augmentation of one vortex of the vortex pair that is typical of symmetric 

JIC’s, and the other is minimized. 

In the JIC model all slots have semicircular ends with their diameter equal to the smaller dimension 

of the slot (which is W), so that L/W = 1 specifies a round hole. The spreadsheet always uses the 

superposition model described in References 5 to 7 when slanted slots are specified; thus, if round holes 

are specified as L/W = 1 slanted slots, it is unnecessary to use the two-row procedure described in 

Reference 7 where the same round-hold configuration was not specified on the top and bottom in any 

row (to avoid invoking the symmetry model). Note that x/H = 0 is at the center of the orifice and that the 

trailing edge of 45 slanted slots is at x/H = 0.5(1+(sin(π/4)(L/W-1))/(H/W). 

For the results shown in this report, the translation function for 45 slanted slots is defined as: 

 z = S sin(a /2) if a < 1 

 z = S if a > 1 

 where a = ( x/H)(((L/W)-1)/1.8)(C/Coptimum)

0.5

 C = (S/H)√J

 Coptimum = 1.25 for opposed rows 

 Coptimum = 2.5 for single-side injection 

The multiplier (L/W-1)/1.8 was added to the translation function for slanted slots so that the 

predicted scalar distributions for L/W = 1 slanted slots (round holes) would not translate in the lateral (z) 

direction, and was normalized for L/W = 2.8 so that previously published empirical model results for 

single-side injection from 45 slanted slots would not change. Translation of 45 slots slanted in either 

direction was also added so the JIC spreadsheet could do perpendicular slanted slot configurations. 
NASA/TM—2011-215980 4 

Results and Discussion 

Figures 2 to 4 show spreadsheet results using the NASA JIC empirical model for opposed rows of 

jets with 45 slanted slots and round holes at conditions used in References 8 (CFD data) and 10 

(experimental data). Schematics and details of the conditions investigated are given in Table 1. 

Figures 2 and 3 show empirical model calculations for the flow and orifice conditions of the 

experiments described in Reference 10. As the development and original publication of the NASA JIC 

empirical model pre-dated these experiments, the only influence the results in Reference 10 had on the 

empirical model were (1) to require that round holes (L/W = 1 slanted slots) must not translate, and (2) 

that the spreadsheet should be able to calculate perpendicular slanted slots which it could not do 

previously. Spreadsheet calculations were performed for opposed rows of 45 slanted slots with DR = 1, 

J = 16, H/d = 8 and Cd = 0.64 at downstream distances of x/H = 0.125 (Fig. 2) and x/H = 0.375 (Fig. 3). 

Figures 2(a) and 3(a) show L/W = 4 slanted slot results from the empirical model for S/H = 0.25 

(SSB in Ref. 10), and Figures 2(b) and 3(b) show results for S/H = 0.5 (SSE in Ref. 10). These are 

parallel inline configurations since the slots on the top and bottom walls have their centerlines aligned 

and the slots are slanted in the same direction. 

The parameter C = (S/H)√J was shown in References 1 et al. to be an excellent indicator of 

penetration for jets in crossflow. For opposed rows of inline jets, optimum penetration occurs for round 

holes when C is approximately 1.25, underpenetration was obvious for C0.6, and overpenetration 

occurs for C  2.5. Thus, for the case of S/H = 0.25 and J = 16 (which gives C = 1), we would expect 

nearly optimum penetration for opposed rows of inline jets at downstream locations. For the case of S/H 

= 0.5 and J = 16 (which gives C = 2), we would expect over penetration for opposed rows of inline jets 

at downstream locations. Increased spacing between orifices (i.e., increasing S/H) results in increased 

penetration. The empirical model should not be used if S/H>1, and note that the jet penetration cannot 

exceed that of an unbounded JIC (infinite spacing and duct height) for the given J. 

The distributions for round holes with S/H = 0.5 (RHD in Ref. 10) are shown in Figures 2(c) and 

3(c). The empirical model distributions for round holes are the same as the slanted slot distributions 

shown in Figures 2(b) and 3(b) except that the distributions for round holes do not translate. 

The configurations in Figures 2(d) and 3(d), 2(e) and 3(e), and 2(f) and 3(f) are respectively, SSF, 

SSG, and SSH in Reference 10. The configuration in parts (d) are L/W = 4 slanted slots at the same 

spacing (S/H = 0.5) as in parts (b) but the slots in parts (d) slant in opposite directions on top and bottom 

walls making a perpendicular inline configuration. The configuration in parts (e) are L/W = 4 slots at the 

same slant direction and spacing as in parts (b) but slots on one side of parts (e) are shifted by S/2 

making a parallel staggered configuration. The configuration in parts (f) are similar to the perpendicular 

slots in parts (d) except that one side of the configuration in parts (f) is shifted by S/2 making a 

perpendicular staggered configuration. 

For distributions with S/H = 0.5, there is virtually no interaction between jets from opposite sides at 

x/H = 0.125, and there is no obvious “best” configuration. Farther downstream at x/H = 0.375, the 

situation is quite different, and the parallel staggered configuration in Figure 3(e) is “best” as one would 

also conclude from perusing the experimental results in Reference 10. Although distributions for 

perpendicular slanted slots are similar to those for parallel staggered configurations at some downstream 

locations, results for perpendicular slanted slots are highly dependent on distance and are no better than 

parallel staggered slots at locations where they are similar and are worse than parallel ones at other 

distances. 

The conditions in Figure 4 are the same as the CFD calculations for opposed rows of inline 45

slanted slots shown in references 1 and 8 (black and white figure in Ref. 1; color in Ref. 8), i.e., 

DR = 2.2, J = 6.6, S/H = 0.5, H/d = 4, and Cd = 1. Note that J is wrong in the title of Figure 12 in 

Reference 8, but is correct in Table 2 therein, and is correct in the title of Figure 29 in Reference 1. Note 

also that the configurations in Reference 1 and 8 are all inline ones and are similar to SSE, SSF, and 

RHD as shown in Table 1 but the aspect ratio (L/W) for the slanted slots is 2.8 in Figure 4(a) and (b) and 
NASA/TM—2011-215980 5 

the equivalent round hole size is H/d = 4. The distributions for 45 slanted slots are shown in Figure 4 at 

x/H = 0.2. C = 1.28 for the conditions in Figure 4 and we would expect nearly optimum penetration for 

inline jets at downstream locations. 

In Figure 4(a) the slots on top and bottom slant in the same direction and distributions translate to the 

right on both top and bottom as is also evident in both the profile and contour plots in References 1 and 8. 

Figure 4(b) shows the same slots in a perpendicular inline arrangement where the distribution from 

the top jets translates to the right and that from the bottom jets translates to the left as do the CFD results 

in References 1 and 8. In perpendicular configurations, the flow translates in opposite directions in the 

two halves of the duct creating the potential for large scale mixing between the halves. However, the 

distributions do not suggest any improvement over parallel slots. Thus, the authors concluded in 

Reference 8 that there does not seem to be any advantage to this configuration, at least at the optimum 

ratio of orifice spacing and momentum-flux ratio for opposed rows of inline round holes. 

For comparison, distributions for inline jets from round holes are shown in Figure 4(c). Note that the 

penetration of round holes and 45 slanted slots is the same in the empirical model, but, as expected, 

distributions for round holes do not translate in either the empirical or numerical calculations. 

A report that includes CFD results for staggered arrangements of 45 slanted slots was published in 

Reference 9. However, it was focused on a large J and large orifice spacing, giving C values that are 

from 5 to 10. Although these C values suggest significant overpenetration and are approximately 5 to 10 

times the values considered here, the authors of Reference 9 also concluded that the parallel staggered 

arrangement was “best” at downstream locations. Even though the JIC spreadsheet permits calculating 

such overpenetration cases, the results are not reported as the vertical profiles would not be consistent 

with the functional form assumed in the JIC empirical model. Profiles that are inconsistent with the form 

assumed in the JIC empirical model are most likely in cases of gross over penetration as the model 

assumes two-sided Gaussian profiles and, for overpenetration, a significant fraction of the profile on the 

side opposite from the injection may be at y/H>1 and if so, the profile is truncated and mass is “lost”. 

Slideshow 

A slideshow showing contour plots from the spreadsheet for one side injection of 45 slanted slots 

has been assembled as Appendix A. The conditions for the sequence shown are the same as in Sequence 

14 in the slideshow in Appendix A of Reference 6 using profile plots. 

Conclusions 

The purpose of this study was to extend the NASA JIC empirical model, implemented in the Excel

spreadsheet documented in NASA/TM—2005-213137, to jets entering the mainstream from opposed 

rows of 45 slanted slots. Results were obtained using a spreadsheet derived from the one posted with 

NASA/TM—2010-216100 with two significant changes that only affect slanted slots: 1) the translation 

function for 45 slanted slots was corrected so distributions for round holes (as L/W = 1 slanted slots) do 

not translate and 2) the capability was added to allow 45 slots to slant in either direction so 

perpendicular slanted slot configurations, where the slots are angled in opposite directions on top and 

bottom walls, could be calculated. 

The empirical model results in this report for opposed rows of 45 slanted slots agree with the trends 

in both the CFD and experimental results published previously and show that parallel slanted slots, 

where the slots are angled in the same direction on top and bottom walls, are the best slanted slot mixing 

configuration at any downstream distance. Although distributions from perpendicular slanted slots are 

similar to those from parallel staggered configurations at particular locations, results for perpendicular 

slot cases are highly dependent on downstream distance and are no better than parallel staggered slots at 

locations where they are similar and are worse than parallel ones at other distances. 
NASA/TM—2011-215980 6 

TABLE 1.—ORIFICE CONFIGURATIONS AND FLOW CONDITIONS 

Figure Code Aspect Orifice H/W3 x/H of J C4

 Ratio spacing trailing 

 L/W S/H edge 

3(c) & 4(c) RHD1

 1 0.5 8 0.063 16 2 

5(c) A2

 1 0.5 4 0.125 6.6 1.28 

3(a) & 4(a) SSB1

 4 0.25 17.6 0.089 16 1 

 

 

3(b) & 4(b) SSE1

 4 0.5 17.6 0.089 16 2 

5(a) E2

 2.8 0.5 6.06 0.188 6.6 1.28 

 

3(d) & 4(d) SSF1 4 0.5 17.6 0.089 16 2 

5(b) F2

 2.8 0.5 6.06 0.188 6.6 1.28 

3(e) & 4(e) SSG1

 4 0.5 17.6 0.089 16 2 

3(f) & 4(f) SSH1

 4 0.5 17.6 0.089 16 2 

1

 in Reference 10 

2

 in Reference 8 

3 H/W = H/d*sqrt(1+4/(L/W-1)) 4 C = (S/H)*sqrt(J) 

 
NASA/TM—2011-215980 7 
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NASA/TM—2011-215980 22 

References 

1. Holdeman, James D. Mixing of Multiple Jets With a Confined Subsonic Crossflow. Prog in Energy 

and Combust Sci, Vol. 19, pp. 31–70, 1993 (similar to AIAA Paper 91-2458 and NASA TM 104412, 

June 1991). 

2. Holdeman, James D., Liscinsky, David S., Oechsle, Victor L., Samuelsen, G. Scott, and Smith, 

Clifford E. Mixing of Multiple Jets With a Confined Subsonic Crossflow: Part I - Cylindrical Ducts. 

J of Eng for Gas Turbines and Power, Vol. 119, No. 10, pp. 852–862, October 1997 (same as ASME 

Paper 96-GT-482 and NASA TM 107185, June 1996). 

3. Holdeman, James D., Liscinsky, David S., and Bain, Daniel B. Mixing of Multiple Jets With a 

Confined Crossflow: Part II—Opposed Rows of Orifices in Rectangular Ducts. J of Eng for Gas 

Turbines and Power, Vol. 121. No. 7, pp. 551–553; July 1999 (same as ASME Paper 97-GT-439 

and NASA TM 107461, June 1997). 

4. Holdeman, James D., Vardakas, Mark A., and Chang, Clarence T. Mixing of Multiple Jets With a 

Confined Subsonic Crossflow: Part III—The Effects of Air Preheat and Number of Orifices on Flow 

and Emissions in an RQL Mixing Section. NASA/TM—2008-215151, March 2008 (similar to J of 

Fluids Eng, Vol. 129, No. 11, pp. 1460–1467, Nov. 2007; derived from NASA TM-1999-209431, 

September 1999). 

5. Holdeman, James D., Smith, Timothy D., Clisset, James R., and Lear, William E. A Spreadsheet for 

the Mixing of a Row of Jets with a Confined Crossflow. NASA/TM—2005-213137, February 2005. 

6. Holdeman, James D., Clisset, James R., Moder, Jeffrey P., and Lear, William E. On the Mixing of 

Single and Opposed Rows of Jets With a Confined Crossflow. NASA/TM—2006-214226, October 

2006. 

7. Holdeman, James D., Clisset, James R., and Moder, Jeffrey P. Spreadsheet Calculations for Jets in 

Crossflow: Opposed Rows of Inline and Staggered Holes and Single and Opposed Rows With 

Alternating Hole Sizes. NASA/TM—2010-216100, July 2010. 

8. Holdeman, J.D., Reynolds, R., and White, C.A. Numerical Study of the Effects of Curvature and 

Convergence on Dilution Jet Mixing. AIAA-87-1953 and NASA TM-89878, June 1987. 

9. Bain, D.B., Smith, C.E., and Holdeman, J.D. CFD Mixing Analysis of Jets Injected From Straight 

and Slanted Slots Into Confined Crossflow in Rectangular Ducts. AIAA-92-3087 and NASA TM105699, July 1992. 

10. Liscinsky, D.S., True B., Vranos A., and Holdeman, J.D. Experimental Study of Cross-Stream 

Mixing in a Rectangular Duct. AIAA-92-3090 and NASA TM-105694, July 1992.