CHAPTER 4: COUPLING TECHNIQUES FOR THE CONJUGATE HEAT
4.3 Hypersonic Flow Over Flat Plate
Fig. 4.2: Flow chart for coupled conjugate heat transfer (CHT) methodology.
Fig. 4.3: Mach contours of fluid domain for hypersonic freestream of Mach number 9 used in DC-CHT
Fig. 4.4: Non-dimensional temperature and velocity profiles at x=0.01 m from the leading edge from DC-CHT.
Further decrease from the maximum value can be seen till the freestream temperature.
It should be noticed here that the temperature profile attains a value inside the thermal boundary layer which is more than the wall temperature (300 K) and the freestream temperature (113 K). Frictional heating in the presence of hypersonic flow is the reason for this temperature enhancement or non-monotonic profile of temperature. This aerodynamic heating is the reason of thicker velocity boundary layer which follows the function relation Eq. (3.28) with the freestream and wall parameters. Therefore, the velocity boundary layer obtained from computations and the one from relation are plotted in Fig. 4.5. Previously encountered agreement in Fig.3.7 has also been observed here. The skin friction coefficient and Stanton number are evaluated from Eq. (3.25) and Eq. (3.27) respectively and are plotted
along with the computed values in Fig. 4.6 and Fig. 4.7 respectively. These figures also confirm the accuracy of present simulations through the encouraging agreement between analytical and computational results.
Fig. 4.5: Variation of velocity boundary layer thickness along the length of plate from DC-CHT
Fig. 4.6: Variation of skin friction coefficient along the length of plate obtained from DC-CHT.
Fig. 4.7: Variation of Stanton number along the length of plate predicted using DC-CHT.
The main objective of present simulations is to conduct the heat transfer analysis between fluid and solid domains in the decoupled manner to evaluate the requirement of active coupling. Heat transfer is obtained analytically using equation below [White,1991],
(
0)
( )s p w
q =St×
ρ
∞×C T −T (4.1) where, T0=T∞(1 0.5+( γ
−1)
M∞2 (4.2) In view of this, the wall heat transfer rate given by present simulations and plotted in, Fig.4.8. This profile forms the boundary condition for solid domain. Macor is considered as the solid material of same length of fluid domain and thickness of 5 mm. Schematic of the solid domain is same as the one shown in Fig. 3.8. During this simulation, initial solid temperature is considered as 300 K while the boundary conditions other than the heat flux are kept same as mentioned in Fig. 3.8.
Fig. 4.8: Heat flux variation along the length of the plate obtained from DC-CHT.
The temperature contours for the solid at different time intervals are shown in Fig.
4.9. Application of non-uniform heat flux (Fig. 4.8) on the top wall has lead to non-uniform temperature variation at all the time instances. Highest temperature of 576.5 K, 740.68 K and 829.22 K has been obtained for the time instances 1s, 5s and 10s. This maximum value has been noticed at the leading edge.
(a)
(b)
(c)
Fig. 4.9: Temperature contours at (a) 1s; (b) 5s; (c) 10s predicted using DC-CHT.
The variation of the fluid-solid interface temperature with time is as shown in Fig.
4.10. This figure also provides the evidence of higher wall temperature at the leading edge and decrement in the same towards the trailing edge. Beyond the 10% of length past the leading edge, uniform temperature of 559.2 K can be noticed at 1s. Thus the varition of temperature is only for first 10% of the length at 1s. However this length corrsponding to temperature variation increases to 20% and 40% at 5s and 10s respectively. This uniform temperature is 259.2 K, 440.685 K and 529.22 K in the increasing order of those time instances. However, this temperature variation has not been communicated to the fluid flow solver in this decoupled or uni-lateral way of heat transfer analysis. Hence the fluid properties and near wall field of the fluid domain, discussed earlier, remain frozen or independent of temperature variation in solid. In view of this, it is evident here that the leading edge non- uniform heat flux distribution would become promiment for experimental heat flux measurement mainly due to observed multidimensionality of heat transfer. Assumption of constant and uniform wall temperature, remain violated for long duration testing.
Fig. 4.10: Temperature distribution along the solid-fluid interface predicted using DC-CHT.
To ascertain this fact, temperature time history at three location on the plate are plotted in Fig. 4.11. Distances of these locations from the leading edge are X1=0.003276 m, X2=0.0107241 m and X3=0.086035 m respectively. This figure re-asserts the earlier observation of higher temperature rise towards the leading edge due to higher heating rates and lower temperature rise downstream due lower heating rates. Temperature time history based on one dimensional heat transfer assumption, given by Eq. (3.20), is also plotted in the same figure based on corresponding heating rates. It is clear here that the one dimensional and two dimensional temperature traces show equivalence for locations X2 and X3. Almost
uniform heating rates in the neighborhoods of these locations are the major reason for those observations. This uniformity of surface heating rates in downstream locations observed in Fig. 4.8 forces one dimensionality. Therefore the temperature rise obtained from one dimensional assumption and two dimensional simulations show good agreement with each other. Moreover at the leading edge, strongly non-uniform heating rates provide sufficient internal temperature gradient along the x-axis which in turn leads to discrimination in temperature traces. In all, these simulations confirm the necessity of active or bi-lateral coupling between solid and fluid domain for accurate prediction of surface variations.
Fig. 4.11: Temperature traces at various locations predicted using DC-CHT.