EXPERIMENTAL INVESTIGATION OF A SUPERSONIC BACKWARD FACING STEP
FLOW
Ferry F.J. Schrijer1and Dario Modenini2
1Delft University of Technology, Kluyverweg 1, 2629 HS, Delft, The Netherlands
2University of Bologna, Italy
ABSTRACT
The flow over a backward facing step is investigated by means of infrared thermography. An inverse heat transfer procedure is used to calculate the surface heat flux from the measured surface temperature in time by taking into account multi-dimensional and unsteady conduction ef-fects. Additionally schlieren visualization is used to get an overview of the flow field. The backward facing step flow is investigated for laminar, transitional and turbu-lent separating boundary layers. It is found that for the laminar and transitional cases the Stanton number and normalized separation length are influenced by the step height. For turbulent separation this is only marginally the case however the Stanton number downstream of reat-tachment increases with step height. For the transitional separation, large heat transfer peaks are measured at reat-tachment.
Key words: Supersonic flow, Transition, Infrared Ther-mography.
1. INTRODUCTION
In high speed aerodynamics backward facing step flows are often encountered. For high-speed vehicles typical examples are the gaps and steps occurring on a tile shaped thermal protection system and other type of irregularities on the vehicle body (control surfaces, junction of different materials). The presence of a step may trigger transition to turbulence causing increased aerodynamic heating which may have implications for the structural integrity of the vehicle. An example is the possible backward or forward facing step on the EXPERT vehicle, the nose of the vehicle is produced out of C/SiC while the afterbody is constructed from metallic parts (PM1000 and Gamma-TiAl). Since both materials have different thermal expansion coefficients, a step may occur when the vehicle surface temperature increases during re-entry. This may cause boundary layer transition to be triggered early during reentry [5].
Figure 1. Occurrence of a backward facing step due to different thermal expansion
Other examples of backward facing step flows are found in scramjets, where step geometries are added next to the injectors to enhance mixing between air and fuel, and plug nozzles where a base flow is created due to trun-cation of the terminal part of the plug. In both cases a proper knowledge of the fluid dynamics phenomena re-lated to step flow geometry is of paramount importance for an accurate estimate of the engine performance. In literature al lot of information can be found on back-ward facing step flows [1], [3], [2], [4] and with this data enough confidence has been established in predicting the base pressure and to correlate the related data using scal-ing parameters, however this is not the case for the heat transfer. A similar conclusion is also valid for the numer-ical codes for which the accuracy of heat transfer rate pre-diction is always less than for the pressure. A particularly controversial point that still remains is the heat transfer overshoot at reattachment, and how this is influenced by the Mach number, Reynolds number and transition. Surface heat flux measurements are classically performed by means of heat flux gauges. In order to get an improved spatial resolution of the surface heat transfer distribution, techniques can be used such as infrared thermography, LCD crystals or TSP. These techniques return the surface temperature, quantitative heat transfer values can be ob-tained by using appropriate data reduction techniques. Of these thermometry techniques, infrared thermography is the most sensitive [8].
Infrared thermography, in combination with a heated thin skin technique, may lead to very accurate heat transfer values [6]. This technique requires a steady state
situa-tion where the surface temperature stays constant and the heat that is generated in the skin is dissipated to the flow. However the time it takes before thermal equilibrium is reached (approx 10 minutes) prevents it to be used in su-personic blow down wind tunnels.
A second approach which is often used is to compute the heat transfer from the surface temperature variation in time. In case when the time duration of the event is short (the model can be regarded as an infinite slab) and the gradients in surface temperature are not too large, the one-dimensional heat conduction equations may be used [7]. However in the current application, the assumptions are not adequate. Therefore the axisymmetric transient heat conduction equations are solved in order to obtain the surface heat flux.
2. INVERSE METHOD
The surface temperature variation in time of the wind tun-nel model is measured by means of an infrared camera. Before the wind tunnel is started, the model is heated by means of radiation (1 kW lamp) to a uniform temperature
of approximately 70oC. When the wind tunnel is started,
the model is cooled down by the flow. The amount of cooling (temperature decrease) is governed by the sur-face heat flux, the model geometry and model material. The objective is to measure the surface heat flux by using an inverse heat transfer technique.
Normally when applying the heat transfer equations, a direct problem is solved where the temperature is calcu-lated as a function of space and time by applying pre-scribed boundary conditions such as spatial temperature distribution, surface heat flux or surface heat transfer
co-efficient. In the application of Inverse Heat Transfer
Problems (IHTP), for example surface temperature mea-surements are used for the determination of the surface heat transfer.
The inverse heat transfer is essentially an optimization process. A surface heat flux variation in time and space is assumed and the surface temperature is calculated by means of a fully implicit finite volume solver, which is compared to the measured surface temperature. In the it-erative process the heat flux is adjusted in such a way that the computed surface temperature approaches the mea-sured temperature. In practise the presence of measure-ment noise may lead to instabilities hence the method is mathematically classified as ill-posed. For the solution to be stable, a regularization technique must be applied to the optimization process. In the current approach the iterative regularization principle is applied which uses a suitable choice of the stopping criteria for the iterations such that the final solution is stabilized with respect to the measurement errors.
The iterative process uses a conjugate gradient method (CGM) that consists of the following parts [9]:
• Adjoint problem; computation of direction of de-scent
• Sensitivity problem; computation of step size in the direction of descent
• Direct problem; computation of the surface temper-ature to check for convergence
In practise the following cost functional is minimized for the computed and measured temperature distribution:
J [q(S, t)] = Z tf t=0 M X m=1 [Tm(t) − Ym(t)] 2 dt, (1)
where Ym(t) is the measured temperature at location m
and time t. Tm(t) is the temperature computed from the
boundary condition ˆq. The CGM consists of the
follow-ing iterative procedure for the heat flux estimate: ˆ
qn+1(S, t) = ˆqn(S, t) − βndn(S, t) (2)
where βn is the search step size obtained through the
adjoint problem and dn(S, t) is the direction of descent
from the sensitivity problem. For a schematic represen-tation of the process, see figure 2. The iterative process is repeated until convergence is reached, in practise this means that the process is stopped when the following cri-terion is satisfied which relates the residual to the mea-surement error [9]:
Jn< σ2M tf (3)
where σ is the standard deviation of the measurement technique, M is the number of spatial points included in
the computation and tfare the number of time steps.
4 Inverse Heat Transfer
72
boundary conditions and time integration, as explained in the mathematical derivation of the method.
Figure 4.5: schematic representation of CGM iterative procedure
One important issue when developing a new computational method is its validation. Therefore CGM is validated using synthetic experiments, as follows. The “measured data” for the numerical experiment is generated by prescribing a certain heat flux matrix to the direct problem. The solution in terms of surface temperature distribution is then used as input for the optimization process, and the converged heat transfer estimate can be compared with the prescribed one.
In this way one can assess the accuracy of the method with respect to different parameters:
- different shapes of the heat transfer distribution; for example smooth variations or sharp discontinuities;
Figure 2. caption
3. EXPERIMENTAL SETUP
The model used in the experiments is an axisymmetric cone-cylinder, see figure 3. A backward facing step is
3 Experimental apparatus
33
3.2.1 Axisymmetric model
The early choice of an axisymmetric model was done for several reasons: it allows a simple manufacturing and the capability to easily change a lot of different step heights, it exerts a reduced blockage effect in the test section, it provides a quasi 2 dimensional flow.
The basic configuration features a cone upstream of the step followed by a cylindrical sting, both being made of steel (see figure 3.3). The maximum step height is determined by the difference between the cone base diameter and the sting diameter. Other step sizes are obtained by means of hollow makrolon sleeves that are mounted on the sting. Makrolon is a synthetic material, easily machinable and capable to withstand service temperature up to 134 °C. The density ! is 1.2 ! 103 Kg/m3, the conductivity k is 0.20 W/(m K) and the
thermal capacity c is 1.17 ! 103 J/(Kg K), resulting in a diffusivity ! = 1.42 ! 10-7 m2/s. Its
emissivity is 0.88.
Figure 3.3: Drawing of the axysimmetric model.
With this configuration step heights of 0, 2, 4, 6, 8, 9, 10 mm are made available. The choice of th is low conductive material is motivated by the application of QIRT, in order to minimize heat losses via conduction.
Two different cones have been manufactured with semi aperture angle of 15 and 25 degrees: this allows to vary the running length upstream of the step, enhancing the capability in changing ReL (since the total pressure of the flow can be varied only by a
factor of almost two). The first model was designed with the steel cone depicted in figure,
Figure 3. Axis symmetric wind tunnel model
obtained by applying a larger base radius to the cone
compared to the cylinder. The model has a stainless
steel core to ensure stiffness and is covered with a low conductive makrolon mantle which has an emissivity coefficient of approximately 0.9. The cone has a base diameter of 30 mm and step heights of h = 2, 4, 6, 8 mm are generated by applying the appropriate mantles. In addition to changing the total pressure, two cones were used in order to change the boundary layer Reynolds number at separation.
Figure 4. ST15 supersonic blowdown wind tunnel The flow facility used in the experiments is ST-15, see figure 4, a supersonic blow down wind tunnel having a
15 × 15 cm2test section. The free stream Mach number
can be set to either 1.5, 2, 2.5 or 3, by using interchange-able sets of liners. In the current investigation, the free stream Mach number is 3. A total pressure of 4.7 or 7.4 bar was used to obtain different unit Reynolds numbers. Since the cone has a leading edge shock and the inves-tigation focusses on the backward facing step, the free stream variables have to be corrected for the extra com-pression due to the leading edge shock. An overview of the flow parameters is given in table 1, it is apparent that the different shock angles do not have a significant influ-ence on the unit Reynolds number. However due to the
extra length of the 15◦cone, the boundary layer Reynolds
number at separation is increased with respect to the 15◦
cone, spanning complementary regions.
A CEDIP Titanium 530L infrared camera was used to ac-quire the surface temperature. The camera has a 320 × 256 pixel Mercury Cadmium Telluride sensor which op-erates in a wavelength band between 7.7 and 9.5 µm.
The measurement accuracy is ±1oC and the sensitivity
is 25 mK. The maximum full frame rate of the cam-era is 250 Hz, however for the current application the acquisition rate was set to 50 Hz. Thermograms were
Table 1. Flow parameters
θ L M∞ Mc Re/m ReL
[deg] [mm] ×10−6[m−1] ×10−6
15 56.6 3 2.6 51 - 81 2.9 - 4.6
25 35.5 3 2.0 51 - 81 1.8 - 2.9
recorded with a spatial resolution of 1 mm/pixel. Since glass is nearly opaque in this wavelength range, optical access is provided by using a Germanium window which has a transmissivity of approximately 0.8. Although the data obtained from the IR camera are planar maps, a line parallel to the model centerline is selected since the in-verse heat transfer process is based on an axisymmetric model; this is consistent with the flow under investiga-tion which is also mostly axisymmetric.
Finally schlieren visualization is used as additional mea-surement technique to visualize the overall flow field such as the leading edge shock, expansion wave at the step, possible lip shock, shear layer and reattachment shock.
4. RESULTS
4.1. Overal flow visualization
In figure 5 the raw thermograms are shows in combina-tion with schlieren images. The leading edge shock is clearly visualized as well as the reattachment shock. The extent of the expansion fan at the location of the back-ward facing step is somewhat harder to discern. Except for the 2 mm step, a lip shock was found in all cases. The separated shear layer was visualized as well, however it could not be inferred if it was laminar or turbulent.
4.2. Transition detection
In order to establish the state of the boundary layer at the location of separation (laminar, tubulent or transitional), experiments were performed aimed at detecting boundary layer transition on the cone. To establish the transition Reynolds number, measurements were performed at dif-ferent unit Reynolds numbers and for two difdif-ferent cones. In figure 6 sample results of the measurements are given. When the boundary layer undergoes transition, the sur-face heat transfer increases and sharp decrease in sursur-face temperature is measured. When transition is detected up-stream of the step (bottom thermogram in figure 6) it is classified as turbulent separation, as transition occurs at the step (upper thermogram in figure 6) the separation is classified as transitional. In case transition is not de-tected the separation is laminar. Furthermore it was found that transition occurs at a constant Reynolds number of
h = 2 mm, M = 3, Re
L
= 2.5 ×10
6
h = 6 mm, M = 3, Re
L
= 4.1 ×10
6
h = 8 mm, M = 3, Re
L
= 4.1 ×10
6
h = 4 mm, M = 3, Re
L
= 2.5 ×10
6
Figure 5. Schlieren visualization and raw thermograms of the flow field (red means higher temperature)
transition
BFS
Figure 6. Temperature maps indicating the transition lo-cation
4.3. Heat transfer profiles
This section discusses the heat transfer profiles as ob-tained from the inverse heat transfer procedure. In the figures the Stanton numbers are given as a function of the streamwise position normalized by the step height (x/h). The origin is placed at the step location and is positive in streamwise position. In all figures the separated region features a low heat transfer region and at reattachment the Stanton number increases after which in downstream direction the Stanton numbers relax again, the turbulent reference level is indicated by the dotted line.
Table 2. State of the separating boundary layer
Re/m [m−1] 25ocone 15ocone
46.6 × 106 laminar transitional 73.3 × 106 transitional turbulent 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 x 10−3 St [ − ] x/h [−] h = 2 mm h = 4 mm h = 6 mm h = 8 mm
Figure 7. 25ocone,Re/m = 46.6 × 106[m−1], laminar
separation
In case of laminar separation the heat transfer is greatly affected by step height h in terms of separation length,
0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 x 10−3 St [ − ] x/h [−] h = 2 mm h = 4 mm h = 6 mm h = 8 mm
Figure 8. 15ocone,Re/m = 73.3 × 106[m−1], turbulent
separation 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 x 10−3 St [ − ] x/h [−] h = 2 mm h = 4 mm h = 6 mm h = 8 mm
Figure 9. 25ocone,Re/m = 73.3 × 106[m−1],
transi-tional separation 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 x 10−3 St [ − ] x/h [−] h = 2 mm h = 4 mm h = 6 mm h = 8 mm
Figure 10. 15ocone,Re/m = 46.6 × 106[m−1],
transi-tional separation
steepness of the heat transfer increase in the reattachment region, and heat transfer peak, see figure 7. This is to be
expected since the step height influences the location of transition when separation is laminar (in the shear layer, during reattachment, or downstream of reattachment). In particular, a sharp peak in St is found for the highest step (8 mm), the magnitude is higher than the turbulent val-ues.
For turbulent separation, figure 8, the heat transfer values are only marginally affected by changing h. The non-dimensional separation length is practically constant for h = 2, 4, 6 furthermore the St increase at reattachment is similar for all cases. A different behavior is found for the largest step height, currently this cannot be explained. It could be attributed to a measurement error.
For a transitional separating boundary layer, measure-ments were done on both cones, figures 9 and 10. The corresponding heat transfer profiles show analogous be-havior in terms of values and profile. Except for h = 2 mm, sharp heat transfer peaks are found at reattach-ment. For h = 8 mm, the highest values are found for all measurement conditions, in this case the heat trans-fer increases up to 1.5 to 2 times the value for obtained for turbulent separation. For a step height of 2 mm a low plateau heat transfer value is obtained. Currently this is cannot be explained, however it is measured for both cones and the measurement is repeatable, therefore the possibility of a measurement error is excluded.
4.4. Separation length
A separation length of xsep/h = 2 − 2.5 was found for
all cases which agrees well with values found in litera-ture. However it was found that on a local scale, bound-ary layer transition has an appreciable effect on the tran-sition length, see figure 11.
Figure 11. Effect of boundary layer transition on the sep-aration length (red means higher temperature)
It was found that when the transition location moves to-ward the step, the separation region becomes smaller.
4.5. Conclusions
The inverse heat transfer technique was successfully ap-plied to a backward facing step flow. The study focussed on the effect of the state of the boundary layer on the reattachment heat transfer profile. Boundary layer tran-sition on the cone upstream of the step was detected by
means of infrared thermography. It was found that for the lowest unit Reynolds number and shortest cone the sep-arating boundary layer was laminar. For the highest unit Reynolds number and shortest cone the boundary layer was transitional at separation, this was also the case for the lowest unit Reynolds number in combination with the longest cone. In case of the highest unit Reynolds num-ber and the longest cone the boundary layer was found to be turbulent at separation. In general it was found that boundary layer transition occurs at a constant Reynolds
number of 1.75 × 106.
In case of laminar separation the flow field downstream of the step was found to be largely influenced by the step height although no clear trend was found. For turbulent separation the step size has a limited effect on the heat transfer values, far downstream of the step the Stanton numbers were found to increase with step height. When the boundary layer is transitional at separation (short cone with high unit Reynolds number and long cone with low unit Reynolds number), high heat transfer peaks at reat-tachment were measured for all step heights except for h = 2 mm.
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