Opti­miza­tion of a Leading Edge for Flat Plate Boundary Layer Experiments

This blog post is part of our CAESES Student Award 2020 com­pe­ti­tion about the use of CAESES in academia, where we showcase exciting material sub­mit­ted to us by students who use CAESES in their research. Many students from high school to PhD make use of CAESES to reach their project goals. If you are one of them, we encour­age you to send us an article about your project and how CAESES has helped you. Inter­est­ing articles will be posted reg­u­larly in our blog, along with some infor­ma­tion about the author, and at the end of February 2021, we will select the best author, who will win some exciting prizes. 

Pre­dict­ing the tran­si­tional boundary layer is still an almost impos­si­ble task for arbi­trary flow con­di­tions. The con­sid­er­a­tion of the impact of boundary con­di­tions such as pressure gradient, free-stream tur­bu­lence and surface rough­ness demand improved tran­si­tion models, which are the interest of a research group at the Karl­sruhe Uni­ver­sity for Applied Sciences.

The Impor­tance of the Leading Edge

In order to inves­ti­gate the tran­si­tion from laminar to tur­bu­lent boundary layer flow, the researchers have built their own wind tunnel, includ­ing a flat plate test section and exchange­able con­toured top and bottom walls. The latter offer the pos­si­bil­ity to generate either generic or real­is­tic (e.g. gas turbine blade) pressure dis­tri­b­u­tions along the flat plate. In the closed-loop wind tunnel, the air coming from the com­pres­sor flows into the settling chamber, where hon­ey­combs and sieves elim­i­nate large tur­bu­lent eddies and homog­e­nize the flow. A further homog­e­niza­tion is reached by the nozzle that accel­er­ates the flow and guides it into the tur­bu­lence gen­er­a­tor, which consists of hor­i­zon­tal and vertical baffle plates. By rotating those plates and adjust­ing the stream­wise position of the grid, the desired tur­bu­lence inten­sity and tur­bu­lent length scale of the flow can be set inde­pen­dently of each other. This offers the pos­si­bil­ity to inves­ti­gate a wide range of flow con­di­tions. The flow then enters the mea­sur­ing section, from where it flows through an air cooler and back to the compressor.

For a long time, the standard in exper­i­men­tal boundary layer inves­ti­ga­tions was to use either sharp or ellip­ti­cally shaped leading edges. It was assumed that the effect of the leading edge on the exper­i­men­tal results measured down­stream is neg­li­gi­ble. In recent years, however, several research groups have shown that the leading edge has a more sig­nif­i­cant impact on boundary layer sta­bil­ity than pre­vi­ously thought. Espe­cially adverse pressure gra­di­ents desta­bi­lize the laminar boundary layer due to an ampli­fi­ca­tion of nat­u­rally occur­ring dis­tur­bances. There­fore, the onset of laminar-tur­bu­lent tran­si­tion moves upstream. The dis­ad­van­tage of ellip­ti­cal leading edges is twofold. Firstly, close down­stream of the edge nose, the pressure gradient is adverse, and secondly, the junction between the leading edge and the flat plate presents a dis­con­ti­nu­ity in cur­va­ture. Both features con­t­a­m­i­nate the exper­i­men­tal results down­stream of the leading edge. In previous exper­i­ments by the research group, the leading edge influ­ence could be min­i­mized by using a nozzle that induces a favor­able pressure gradient along the leading edge and the sub­se­quent flat plate. The focus of further research projects, however, is on unac­cel­er­ated free-stream flows. This demands an opti­mized leading edge, where the optimum is char­ac­ter­ized as a pressure gradient that is zero as far upstream as possible and nowhere adverse. In order to evaluate the flow behavior of a leading edge, the dimen­sion­less pressure coef­fi­cient 𝐶𝑝 is intro­duced as

Opti­miz­ing the Contour

CAESES was intro­duced to optimize the leading edge contour. It is designed as a B‑Spline with 11 control points, while the result­ing geometry is exported as an STL-file. The mul­ti­di­men­sional opti­miza­tion starts with a global search for an optimal leading edge using the MOSA (Multi-Objec­tive Sim­u­lated Anneal­ing) algo­rithm, followed by a local search with TSearch algo­rithm. The length of the leading edge 𝑎 is defined as the distance from the leading edge nose to the junction with the flat plate. The thick­ness 𝑑 is the flat plate thickness.

Geo­met­ric Constraints

The researchers intend to run hot film sensor mea­sure­ments along the flat plate surface at arbi­trary stream­wise posi­tions. The hot film sensor is mounted on a movable steel belt (30 μm thick), which sur­rounds the flat plate includ­ing the leading edge. A stepper motor drags the steel belt around the plate to adjust the stream­wise position of the hot film sensor. To ensure an undis­turbed flow over the leading edge, the belt needs to fit tightly around the leading edge contour. Unfor­tu­nately, a sharp leading edge would deform the belt per­ma­nently. Con­se­quently, the maximal cur­va­ture of the leading edge was limited by a geo­met­ric con­straint. A maximal cur­va­ture is con­verted to a minimal radius of cur­va­ture, whose cor­re­spond­ing circle is illus­trated in the fol­low­ing figure. Pre­lim­i­nary exper­i­ments have shown that a minimum radius of cur­va­ture of approx­i­mately 2.5 mm should prevent plastic defor­ma­tion of the steel belt.

CFD Coupling with OpenFOAM

The researchers coupled CAESES with the open source CFD library OpenFOAM. To ensure parallel stream lines along the flat plate, the stag­na­tion point has to be at the leading edge nose. By increas­ing the outlet pressure above the plate, while the outlet pressure below the plate remains unaf­fected, the position of the stag­na­tion point can be adjusted towards the top of the leading edge. It can be moved down­wards by decreas­ing the upper outlet pressure. An opti­miza­tion process ensures the position of the stag­na­tion point to be at the leading edge nose. The result­ing pressure contours as well as the stag­na­tion point are shown below.

Results

In order to clarify the improve­ments achieved by the opti­miza­tion method, the opti­mized leading edge contour is compared with two ellipses. The first one is a 12:1 ellipse, which means its semi-major axis is 12 times longer than the semi-minor axis. It has the same length 𝑎 as the opti­mized plate contour. It does not fulfill the geo­met­ric con­straint described above, since the minimal radius of cur­va­ture is too small (≈ 1.0 mm). The second ellipse is a 5:1 ellipse and is sig­nif­i­cantly shorter, but fulfills the geo­met­ric con­straint with a minimal radius of cur­va­ture of 2.5 mm.

The fol­low­ing figure shows the dimen­sion­less pressure dis­tri­b­u­tion for all three edge contours. For the opti­mized leading edge, the pressure gradient is zero and constant at about 40% of the edge length 𝑎. The small adverse pressure gradient down­stream of the nose is neg­li­gi­ble compared to that of the ellipses, as it occurs only close to the nose. Con­trar­ily, for both ellipses, the pressure gradient is non-zero well down­stream of the edge nose, indi­cated by a rising pressure coef­fi­cient 𝐶𝑝. For the 5:1 ellipse, 𝐶𝑝 has a minimum at approx­i­mately 𝑥/𝑎 = 0.2, result­ing in an adverse pressure gradient between 𝑥/𝑎 = 0.2 and 𝑥/𝑎 = 1.0. Although the 12:1 ellipse induces a much smoother 𝐶𝑝 dis­tri­b­u­tion, it still has a minimum even further down­stream at approx­i­mately 𝑥/𝑎 = 0.8. This results in an adverse pressure gradient until the end of the sim­u­la­tion domain at 𝑥/𝑎 = 1.2 that may reach far into the mea­sure­ment section. Although the adverse pressure gradient is quite small, this would result in decel­er­ated free-stream flow, affect­ing the exper­i­men­tal results.

The opti­miza­tion process is shown in the fol­low­ing ani­ma­tion, where both the vari­a­tion of the contour and the result­ing pressure coef­fi­cient dis­tri­b­u­tion are dis­played. It shows clearly that even small vari­a­tions in the contour can highly affect the result­ing pressure coef­fi­cient. This requires a high man­u­fac­tur­ing accuracy during the pro­duc­tion of the opti­mized leading edges.

Con­clu­sion

CAESES helped opti­miz­ing the contour of the leading edge of a flat plate by varying the control points of the B‑Spline contour. The dis­tri­b­u­tion of the pressure coef­fi­cient 𝐶𝑝 served as an objec­tive function, such that it reached a constant value of zero as far upstream as possible. The pressure gradient is only slightly adverse directly down­stream of the opti­mized leading edge nose. The geo­met­ric con­straint of a minimal radius of cur­va­ture probably prevents a leading edge with a more ideal pressure gradient dis­tri­b­u­tion. Compared to the two ellipses, the new design of the leading edge is clearly superior.

About the Author

Thanks a lot to Philipp Masino and his col­leagues from Karl­sruhe Uni­ver­sity of Applied Sciences, for sub­mit­ting this inter­est­ing article and images about their research and opti­miza­tion with CAESES.