Balance of energy

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -176- The decrease in C*_mom,y at the lower boundary of the jet is about 65% and 78% for M =1 and M=1.25 at St=0.2 compared to steady blowing. At downstream location of x/d=5.5, an overall reduction of C*_mom,y in the regions of pertinent velocity gradient seems to occur much rapidly.

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -177- dy

q v d

T_k,y =1/2. ( . ²)/

o subterms of pressure transport

dx p u

x d

k_, 1/ρ. ( . )/ π =

dy p v

y d

k_, 1/ρ. ( . )/ π =

o subterms of production

(

)

uv

P_k,sh = / + / dy V d v dx U d u

P_k,nor = ². / + ². /

o term of rate of dissipation

. _z2

k νω

ε =

In the next section, different terms and subterms of energy equation are presented after normalization. A star symbols used as a superscript over the symbol of any particular term indicates that the term is normalized by using U_∞³ /d.

5.7.3.1 Terms of Energy equation, time-averaged results

The profiles of different quantities of energy equation are shown in figure (5.34), for a steady blowing condition of M = 1. Figure contains the profiles of the subterms of convection corresponding to the spatial derivative in each direction (C*_k,x, C*_k,y) and the total convection C*_k, as well as the subterms of production that are multiple of turbulent shear stress (P*_k,sh) and the subterms of production that are multiple of turbulent normal stress (P*_k,nor) along with the total production (P*_k). The profiles are obtained from a downstream location of x/d=1.5 and z/d=0. Figure (5.34) shows that the peak values of different convection term lie at a similar distance from the wall, while the peak of the total production term, P*_k, lies at a slightly lower height. Higher wall-normal location of the peak of convection term is possibly due to the mean velocities, which are the multiplier of the convection term, and are maximum in the internal region of the jet flow. There is also a slight difference in the peak value of the production subterms pertaining to the normal and shear stresses. It shows that P*_k,sh is higher in the lower part of mixing layer region. The peak of P*_k,nor seems to correspond better to the location of lower boundary of the jet specified from the criteria of vorticity. It should be noticed that the peak of P*_k,nor is higher than P*_k,sh. Andreopoulos and Rodi (1984) have also observed such trends in case of a vertical jet issuing into a cross-stream flow with a blowing ratio of M= 0.5. They particularly highlight the zone near the upstream edge of the hole exit for such results. They reasoned out that having

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -178- dz

W

d / an order of magnitude higher than the dU /dy and dV /dx produces considerable turbulent energy due to the lateral divergence of the flow in the upstream part of the edge of the exit. For the profiles shown in figure (5.34), the contribution of the P*_k,nor to the overall production in the peak region is 62%. Furthermore, it can be noticed that the normal- stress production have negative value below the jet due to dU /dx, see figure (5.31b).

Results of Andreopoulos and Rodi (1984) also indicate a small region of negative production for a profile lying at x/d=4 and z/d=0.

Figure (5. 34): Profiles of the subterms of energy equation for M= 1 and St=0, lying at a location of x/d=1.5 and z/d=0

Due to strong three-dimensional characteristics of the flow, it is hard to characterize the relative importance of the normal- and shear-stress production in different regions of the flow. The energy balance shown in the present study, only gives a tentative picture of the energy transport, as the data in the third dimension of the flow is not available. A qualitative comparison of the present results with Andreopoulos and Rodi (1984), and Muppidi and Mahesh (2007) is not always consistent because of the lack of information of the flow in the third dimension, as well as the difference in the configuration of injectant flow, since the jet is issuing perpendicularly to the mainstream flow in the given references.

The energy equation applied to the plane of symmetry (z/d=0) by Andreopoulos and Rodi (1984) includes an additional term for normal stress production (w²−v²).dW /dz^compared to the present study, which produces significant energy near the upstream edge of the hole in their case. Moreover, to obtain the energy dissipation rate, they used the spectra of velocity fluctuation, as well as the difference of the other terms to compare the results of dissipation rate. However, in the present case, the rate of dissipation was determined by calculating the root-mean-square of the vorticity in the lateral direction of the flow, ε_k^.

In order to look at the dominance of the shear-stress production as was mentioned earlier, the percentage of normal-stress production with respect to the shear-stress production is shown in figure (5.35a) and (5.35b). It can be observed that the region

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -179- containing dominant production due to shear-stress locating near the upstream edge of the jet and the mixing layer region lying at the bottom of the jet seems comparatively wider for the lower blowing ratio (M = 0.65). The regions with dominant P*_k,nor are actually blanked in the figures.

(a) (b)

Figure (5. 35): Percentage of normal-stress production with respect to the shear-stress production in steady blowing (St=0), (a) M= 0.65 and (b) M= 1.

The profiles of different terms of energy equation are shown in figure (5.36a-f), for the steady blowing of M = 0.65, 1 and 1.25 (M increasing row-wise in the array of subfigures).

Figure (5.36a), (5.36c) and (5.36e) show the profile extracted from y/d=0.2 and z/d=0. The profiles extracted from the near-field region of x/d=1.5 and z/d=0 are shown in Figure (5.36b), (5.36d) and (5.36f). For M = 0.65, the profiles shown in Figure (5.36a) and (5.36b) indicate that the role of the turbulent transport is considerably small compared to the other terms. The peak in horizontal profile locating near the upstream edge of the hole corresponds to the leading edge boundary of the jet. On the lee side of the hole, the production of energy dominates and the transport of energy due to the convection term stays comparatively low. The maximum of the production of turbulent kinetic energy always locates near the lee side of the hole in all cases. Andreopoulos and Rodi (1984) has shown that at the downstream region of x/d=4 the shear-stress production is considerably higher than the normal stress, for M = 0.5. However in the present results, all these term decays quite rapidly even at the downstream location of x/d=3.5. The peak value of the total production at x/d=3.5 is only 14% of the peak value measured at x/d=1.5. At the downstream location of x/d=1.5, the shear-stress production is slightly higher than the normal-stress production, and the contribution of the P*k,sh to the overall production in the peak region is 60%. For M = 1, the profiles shown in Figure (5.36c) and (5.36d), as well as for M = 1.25, shown in Figure (5.36e) and (5.36f) indicate that the convection is bit intermittent at the jet boundary lying near the upstream edge of the hole. At the downstream location of x/d=1.5, the peak value of the normal-stress production is higher than the peak value of

“Experimental aerothermal characterization of a pulsating jet issuing in a crossflow:

Influence of Strouhal number excitation on film cooling” -180- shear-stress production, because of dU /dyfdU /dx ^and uv.dU /dypu².dU /dx^. However, at the lower side of the mixing layer, the shear-stress production is higher than the normal-stress production. The contribution of the P*_k,sh to the overall production in the peak region is 38% and 26% for M = 1 and 1.25 respectively. Moreover, in the mixing layer region, the total production is roughly about 2.4 times the dissipation, compared to 1.5 mentioned by Andreopoulos and Rodi (1984). With different approximations mentioned above, a rough estimate for (w²−v²).dW /dz was found to be 5%, 10% and 10% of the peak value of normal-stress production for M = 0.65, 1 and 1.25 respectively, at a streamwise location of x/d=1.5.

The qualitative similarities found in the behavior of the present flow and the reference flow are the appearance of peak of production and dissipation term at the upstream edge of the jet and at the region of mixing layer lying at the bottom of the jet, as well as the appearance of higher normal-stress production compared to the shear-stress production at various regions (see figure (5.35) for details). One of the inconsistencies between the present results and the reference (Muppidi and Mahesh 2007) is the role of turbulent transport, which appears significantly small compared to the other terms in the present study. In the reference article, this term is shown to be significant on the center streamlines (such as; x/d=0.11 and y/d=3.22, with origin at the hole centre) and less important on the jet edge, while employing a vertical jet and a blowing ratio of 5.7 in his study.

(a) (b)