American Institute of Aeronautics and Astronautics

1

Analysis and Modeling of Flow through Wind Tunnel Porous Wall

Taisuke Nambu1

Graduate School of Fundamental Science and Engineering, Waseda University, Shinjuku, Tokyo, 169-8555, JAPAN

Atsushi Hashimoto2 and Takashi Aoyama3

Japan Aerospace Exploration Agency (JAXA), Chofu, Tokyo, 182-8522, JAPAN

and

Tetsuya Sato4

Graduate School of Fundamental Science and Engineering, Waseda University, Shinjuku, Tokyo, 169-8555, JAPAN

A porous wall model for wind tunnel analysis has been newly developed. To establish the model, flows through a hole were directly analyzed by a CFD method. The flow through the hole was computed in different flow conditions and hole shapes. From these computations, we formulated a linear relation between the flow rate and the differential pressure, including the effects of hole size and boundary layer thickness. This linear model is partially validated by an experiment. Then, we compared the computation result of a porous wall using the present model with that by a direct computation. It suggests that the present model has enough accuracy for the simulations of wind tunnel with porous walls.

Nomenclature 𝑥,𝑦, 𝑧 = coordinate 𝑢, 𝑣,𝑤 = velocity of x,y,z direction 𝑝 = pressure 𝑝𝑡 = total pressure ∆𝑝 = differential pressure ∆𝐶𝑝 = differential pressure coefficient 𝑇𝑡 = total temperature 𝑚 = mass flow rate 𝑚′ = nondimensional mass flow rate 𝐹𝑓 = viscosity loss 𝜈 = kinematic viscosity coefficient 𝐿 = depth of hole 𝐷 = diameter of hole 𝛼 = porosity 𝛿∗ = displacement thickness of boundary layer 𝐶𝑓 = friction coefficient 𝑅𝑒 = Reynolds number 𝑅𝑒𝑥 = local Reynolds number 𝐶𝑑 = discharge coefficient 𝑀 = Mach number 1 Graduate Student, Department of Applied Mechanics, AIAA Student Member. 2 Researcher, Aerospace Research and Development Directorate, Numerical Analysis Group, AIAA senior member. 3 Section Leader, Aerospace Research and Development Directorate, Numerical Analysis Group, AIAA senior member. 4 Professor, Department of Applied Mechanics, AIAA senior Member.

40th Fluid Dynamics Conference and Exhibit28 June - 1 July 2010, Chicago, Illinois

AIAA 2010-4858

Copyright © 2010 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute of Aeronautics and Astronautics

2

Subscripts ∞ = value of free stream 𝑝𝑙𝑒𝑛𝑢𝑚 = value in plenum chamber 𝑤𝑎𝑙𝑙 = value on wall

I. Introduction igh accuracy is needed for wind tunnel testing to design aircrafts. Since even a

slight drag reduction contributes to the improvement of aerodynamic performance, aircraft makers desire highly precise measurement (error of under 1 count in drag coefficient). However, such high precision has not been achieved yet despite many efforts for long years because for more reliable measurement, further investigations are needed. In Japan, JAXA 2m x 2m transonic wind tunnel (JTWT) has been often used to investigate the performance of aircrafts. So far, the measurement accuracy of JTWT is improved by the studies such as support interference correction1, flow quality measurement2,3, and thermal zero-shift correction of balance4, In several error sources, wall and support interference is a large issue. Since there is no wall or support in the actual flight, these interferences cause difference between a flow of actual flight and that in wind tunnel. Therefore, correction methods of these interferences are essential to predict the aerodynamic performance. For the support device interference, some correction methods are proposed5, 6. On the other hand, the Mokry's method7 and the panel method8 are used to correct the wall interference for JTWT. However these existing methods need to be improved to get high precision data because there are some unconsidered factor, for example mutual interference between the wall and the support effects. For better correction, we need to investigate the flow inside the wind tunnel in detail and know how the wall and the support devices affect the flow. Computational Fluid Dynamics (CFD) helps us to solve the above problems. Hashimoto et al.9 analyzed a whole configuration of JTWT including an aircraft model, walls, and support devices by a CFD method (see Fig. 1). We can know the difference caused by the interference from the comparison of this result with a free flight condition (without wall and support devices). In the whole wind tunnel analysis, it is important to create a virtual wind tunnel similar to the actual one as much as possible. Generically transonic wind tunnels have porous walls or slotted walls at their test section in order to alleviate the blockage effect caused by test models. In this analysis, the porous walls of JTWT are modeled as a boundary condition based on the modified Harloff’s model12, 13 instead of making an enormous effort on simulating each small hole on the walls. However the predicted pressure distributions on the walls do not correspond well with experimental data9. One of the reasons is that the modified Harloff’s model, which was designed to be applied for the bleed holes of supersonic air-intake10-13, is not always proper for the flow through a wind tunnel porous wall, where differential pressure between the both sides of the wall is small. The existing porous wall models like the modified Harloff’s model are commonly used under the conditions where the differential pressure is approximately 10-100 times larger than that of JTWT. Consequently the purpose of this paper is to propose a new porous wall model for the small differential pressure like in the transonic wind tunnel.

H

Figure 1. Whole wind tunnel analysis9.

American Institute of Aeronautics and Astronautics

3

II. Analysis of flow through single hole We take a strategy of simulation for flow

through single hole by CFD in order to establish a porous wall model in small differential pressure because a previous study shows modeling of porous wall is possible by extending a model of a flow through one hole12.

Figure 2 shows a schematic of the analysis. The primary concern is the flow rate through a hole. We compute the flow rate with changing the differential pressure between the entrance (wind tunnel side) and the exit (plenum chamber side) of hole. The analysis is conducted as follows

1) The flow rate is computed with changing the differential pressure by the same flow condition and hole's shape as the aircraft wind tunnel testing of JTWT (𝑀∞ = 0.8, 𝐿 = 𝐷 = 12𝑚𝑚)

2) The flow rate is computed with changing free stream Mach numbers (𝑀∞ = 0.5 − 0.9)

3) To create a more wide-range model, Effects of hole shape, deepness L and diameter D, are investigated (L/D)

4) The boundary layer conditions may affect the flow strongly, therefore effect of boundary layer thickness is investigated (𝛿∗/𝐷)

The different pressures are set between the wind tunnel and the plenum chamber sides, and then flow rates are computed. The flow field is different in the outflow (wind tunnel to plenum chamber) or inflow (plenum chamber to wind tunnel) cases, and therefore both cases are investigated.

III. Computational methods and grids The following section outlines the computational methods and grids.

A. Computational methods In all computation, FaSTAR (FAST Aerodynamic

Routines) code developed by JAXA is used as a flow solver. In FaSTAR, full Navier-Stokes equations are solved on unstructured grid by a cell centered finite volume method. The Roe's method is used for the numerical flux computation. The LU-SGS (Lower/Upper Symmetric Gauss-Seidel ) implicit method is applied for time integration. The time integration is carried out by the local time stepping. The second-order spatial accuracy is realized by a linear reconstruction of the primitive variables with the Venkatakrishnan's limiter and the MUSCL scheme. As for the turbulence model, the Spalart-Allmaras model is used.

B. Computational grid Figure 3 shows a computational grid and boundary

conditions. The deepness, L, and diameter, D, of the each hole on the porous walls of JTWT are 12mm and 12mm, respectively. The values of L and D are changed in the sections 7 and 8 (effects of L/D and 𝛿∗/𝐷). The

Figure 2. Outline of CFD analysis.

Figure 3. Computational grid and boundary conditions.

American Institute of Aeronautics and Astronautics

4

grid consists of three domains, wind tunnel side, hole, and plenum chamber side. The total number of cells is approximately 1.0 - 2.0 millions. The density of grid is higher near the hole than that of other domains. The grid is longer in the upstream direction to compute the growing boundary layer, and this upstream length is estimated from the result of the whole wind tunnel analysis9. The wall friction coefficient 𝐶𝑓 on the porous wall is obtained from whole wind tunnel analysis and the local Reynolds number 𝑅𝑒𝑥 is calculated from the following equation14

𝐶𝑓 =0.455

𝑙𝑛2(0.06𝑅𝑒𝑥) (1)

As a result, 𝑅𝑒𝑥 is 8.1 × 107 and the upstream length 𝑥𝑢𝑝 is about 550𝐷 (𝐷 = 12mm) from the following equation.

𝑥𝑢𝑝 =𝑅𝑒𝑥𝜈𝑢∞

(2)

IV. Effect of differential pressure Figure 4 shows the contours of Mach number on the cross section of the symmetric plane. Four types of flow are

shown (outflow or inflow, large or small differential pressure). The coefficient ∆𝐶𝑝 is defined as follows

∆𝐶𝑝 =𝑃∞ − 𝑃𝑝𝑙𝑒𝑛𝑢𝑚

0.5𝜌∞𝑢∞2 (3)

In the outflow cases (Figs. 4(a) and (b)), there is large separation inside the hole and this separation narrows the flow passage. Meanwhile in the inflow cases (Figs. 4(c) and (d)), the flow passage is bent by the wind tunnel flow at the hole exit although there is no large separation in the hole. In these ways, the wind tunnel flow decreases the flow rate in both inflow and outflow cases. Under the small differential pressure, these effects of wind tunnel flow are strong as shown in Figs. 4(a) and (c). However as differential pressure increases, these effects become relatively small because of the increase of flow rate through the hole. The flow direction and the range of differential pressure cause large difference. Consequently, the flow field should be investigated separately in each flow type.

Figure 4. Contours of Mach number (𝑴∞ = 𝟎.𝟖).

American Institute of Aeronautics and Astronautics

5

In the first stage of general modeling for porous walls, the shape of hole is fixed as that used in JTWT and a typical flow condition in JTWT is chosen as follows: 𝑀∞ = 0.8, 𝑝𝑡 = 100 [kPa],𝑇𝑡 = 320 [K].

A. Relations between flow rates and differential pressures Figure 5 shows the relation between the flow rate 𝑚′ and the differential pressure ∆𝐶𝑝 under the large differential pressure (∆𝐶𝑝 < −0.1, 0.1 < ∆𝐶𝑝). The variables are nondimensionalized by the values of the wind tunnel flow. The flow rate per unit of time and volume is nondimensionalized as the following equation

𝑚′ =(𝜌𝑣)𝑝𝑜𝑟𝑜𝑢𝑠

(𝜌𝑣)∞ (4)

In this range of differential pressure, the modified Harloff’s model is reliable. Therefore to validate the present computation, the result is compared with that of modified Harloff’s model. The modified Harloff’s model is expressed in Table.1,

The result shows nonlinear relation similar to the modified Harloff’s model. This result validates the present computation although some discrepancies are observed because the boundary layer condition is unconsidered in the modified Harloff’s model. On the other hand, the present result under the small differential pressure (−0.1 ≤∆𝐶𝑝 ≤ 0.1) shows linear relation and large difference exists between it and that by the modified Harloff’s model as shown in Fig. 6.

Table 1. Modified Harloff’s model Modified Harloff's model

Base line (𝜌𝑈)𝑝𝑜𝑟𝑜𝑢𝑠 = 𝐶𝑑𝐻 ∙ 𝜌𝑤𝑎𝑙𝑙 ∙ 𝑈ℎ𝑜𝑙𝑒 ∙ 𝛼

0.1 ≤ ∆𝐶𝑝 ∆𝐶𝑝 ≤ −0.1

Discharge coefficient (0.1 ≤ ∆𝐶𝑝)

𝐶𝑑𝐻 = 𝐶𝑑𝐵 + ∆𝐶𝐷𝑃 + ∆𝐶𝐷𝜃 + ∆𝐶𝐷𝐿𝐷

⎩⎪⎪⎨

⎪⎪⎧𝐶𝑑𝐵 = 0.9800 − 0.2077𝑃𝑟 + 1.4526𝑃𝑟2

−4.6701𝑃𝑟3 + 5.1608𝑃𝑟4 − 1.8960𝑃𝑟5

∆𝐶𝐷𝑃 = −0.46(𝑃𝑟 − 0.5)𝐶𝐷𝜃 = −0.06 − 0.4(𝑀∞ − 0.6)

∆𝐶𝐷𝐿/𝐷 = 0.08

𝑃𝑟 =𝑝𝑝𝑙𝑒𝑛𝑢𝑚𝑝𝑤𝑎𝑙𝑙

�

Discharge coefficient (∆𝐶𝑝 ≤ −0.1)

𝐶𝑑𝐻 = 0.2759∆𝐶𝑝3 − 1.0613∆𝐶𝑝2+ 1.5808∆𝐶𝑝 − 0.0074

Theoretical velocity 𝑈ℎ𝑜𝑙𝑒 = � 2𝛾𝛾 − 1

𝑝𝑤𝑎𝑙𝑙𝜌𝑤𝑎𝑙𝑙

�1 − �𝑝𝑝𝑙𝑒𝑛𝑢𝑚𝑝𝑤𝑎𝑙𝑙

�𝛾−1𝛾�

American Institute of Aeronautics and Astronautics

6

B. Comparing with experiment For the validation of the present computation in small differential pressure, an experiment is conducted in a collaborative research between JAXA and Nagoya University. Figure 7 shows the schematic of the experiment. The experiment employs almost the same conditions as the computation. Figure 8 shows the comparison between the experiment and the present computation. The experiment data shows the same linear relation with the computation. Both gradients are almost same (experiment: 1.05, and computation: 1.10). However there is quantitative difference between two results. Probably pressure distribution over the wall model causes measurement errors of the free stream pressure. Although there is a slight difference in the gradient, this result successfully validates the present computation.

Figure 7. Schematic of experiment15.

Figure 5. Variation of mass flow rate (large

differential pressure).

Figure 6. Variation of mass flow rate (small

differential pressure).

American Institute of Aeronautics and Astronautics

7

C. Modeling the relation between flow rates and differential pressures The present computation suggests the flow field changes significantly as the differential pressure becomes large (Fig. 5 and 6). The linear relation between the flow rate and the differential pressure becomes the nonlinear relation. If the flow field changes significantly, the method of modeling should be changed.

The modified Harloff’s model is based on the theory of nozzle (the discharge coefficient and the theoretical flow rate: see table 1). This theory is derived from the isentropic hypothesis. However, this assumption is not assured under the small differential pressure. In order to examine whether the flow is isentropic, momentum balance expressed as follows is computed inside the hole

∆𝑝 = (𝑚𝑣)𝑜𝑢𝑡 − (𝑚𝑣)𝑖𝑛 + 𝐹𝑓 (5) where 𝑚𝑣̇ is momentum advection and 𝐹𝑓 is viscosity loss. Figure 9 shows the ratio between the change of momentum advection and viscosity loss. As the differential pressure becomes small, the ratio of viscosity loss is large. It suggests the isentropic theory would not be appropriate, in other words, the modeling based on nozzle’s theory does not fit the flow under the small differential pressure.

Figure 8. Comparison of the computation and the experiment.

American Institute of Aeronautics and Astronautics

8

The linear relation is quite characteristic of the flow under small differential pressure. This relation is similar to that in the Hagen-Poiseuille flow. The theory of Hagen-Poiseuille flow says that the relation between the flow rate and the differential pressure in a flow through a pipe is linear like the present computation if the flow field has large viscous force (𝐹𝑓) and small change in momentum advection ((𝑚𝑣)𝑜𝑢𝑡 − (𝑚𝑣)𝑖𝑛). The analysis of momentum balance shows the similar tendency as the differential pressure becomes small. We computed the momentum balance inside the hole, and then the analogy with two flow theories (nozzle’s theory and Hagen-Poiseuille flow) was investigated. From these analyses, we suggest two modeling methods according to the differential pressure. a) Under the small differential pressure (like wind tunnel porous wall) The present computation shows the linear relation between the flow rate and the differential pressure. The experiment also indicates the same linear relation. This relation is analogy with the theory of Hagen-Poiseuille flow and the flow field of present computation turns out to be similar to Hagen-Poiseuille flow as the differential pressure is small. Consequently we model the flow under the small differential pressure as following equation

(𝜌𝑣)𝑝𝑜𝑟𝑜𝑢𝑠(𝜌𝑣)∞

= 𝐴 ∙ 𝛼 ∙ ∆𝐶𝑝 (6)

where A is a constant number and 𝛼 is porosity (ratio of void in the wall). b) Under the large differential pressure (like bleed system of air intake) For the large differential pressure, several porous models already exist. Most of them are based on the nozzle’s theory adding the effect of free stream. For example, the modified Harloff’s model is described as following equation

(𝜌𝑣)𝑝𝑜𝑟𝑜𝑢𝑠 = 𝐶𝑑𝐻 ∙ 𝜌𝑤𝑎𝑙𝑙 ∙ 𝑈ℎ𝑜𝑙𝑒 ∙ 𝛼 (7)

Figure 9. Ratio of between the change of momentum advection and viscosity loss (𝑴∞ = 𝟎.𝟖).

American Institute of Aeronautics and Astronautics

9

where 𝑈ℎ𝑜𝑙𝑒 is theoretical velocity of nozzle’s equation and 𝐶𝑑𝐻 is the Harloff’s discharge coefficient (see table 1). In the present computation, the analysis of momentum balance shows the flow is nearly isentropic as the differential pressure is large. In addition, the effect of free stream becomes relatively small, because the flow rate through the hole is large. Consequently the modeling method based on nozzle’s theory is suitable from the aspect of the flow characteristics.

V. Effect of free stream Mach number In the previous section, we suggested the linear model obtained from the computational results in Fig.6. However the flow rate is computed only at 𝑀∞ = 0.8. In this section, we investigate the effect of free stream Mach number on the flow. We mainly deal with the subsonic conditions (𝑀∞ = 0.5 − 0.9), which is the range of cruise speed for commercial aircraft. Figure 10 shows the relations between the flow rate and the differential pressure for each free stream Mach number. All of the relations are almost identical. This result suggests that nondimensionalization (Eq. 6) is effective. As above, the constant number 𝐴 in Eq. 6 is common for the Mach numbers in the present condition. Therefore, the value of 𝐴 can be specified as follows for JTWT.

𝐴 = 1.01 ( 𝐹𝑜𝑟 𝐽𝑇𝑊𝑇) (8) The porous wall of JTWT can be modeled by Eq. 6 and 8.

VI. Effect of hole’s deepness and diameter In order to extend the above porous wall model for more general purposes, we consider the ratio of hole depth L to diameter D (L/D) in this section. Computational conditions are same as the previous section, except for the computational grid. To compute the flow with various L/D, we make the grid in the hole domain according to L/D.

A. Change in the flow field by L/D effect Figure 11 shows the relation between the differential pressure and the flow rate in various hole shapes. In the inflow cases, the flow rate as a function of differential pressure is almost same with the various L/D, whereas in outflow cases, the flow rate increases as L/D becomes large. At first, we investigate the outflow cases. To investigate the details of flow field, the velocity component of hole axis is visualized with each L/D in Fig. 12. This figure shows only the region of 𝑣>0 by color contours. The flow passage becomes narrow by the separation near the entrance (1~2 in Fig.12). Then the separation area becomes small near the hole exit (2~3 in Fig.12). The separation area at the exit is significantly different, depending on L/D. and it changes the flow rate. In order to quantitatively understand the change of flow field with L/D, Fig. 13 shows the distributions of the velocity component and the pressure coefficient along the sampling line shown in the upper sketch in this figure. These flow passages inside the hole are similar to the flow through the Laval nozzle, which nozzle has different exit area and length. On the other hand, in inflow cases the flow rates are almost same because there is no large separation inside the hole (Fig. 4) and the flow field does not change in the direction of hole axially. Therefore flow passages are not affected by the hole depth like the outflow cases.

Figure 10. Variation of mass flow rate (𝑴∞ = 𝟎.𝟓 −𝟎.𝟗).

Figure 11. Variation of mass flow rate (L/D effect, 𝑴∞ = 𝟎.𝟖).

American Institute of Aeronautics and Astronautics

10

Figure 14. Variations of 𝑭𝑳𝑫

Figure 12. Contours of hole axis velocity (𝑴∞ =

𝟎.𝟖,∆𝑪𝒑 = 𝟎.𝟎𝟓 visualization only the region of 𝒗>0).

Figure 13. Variations of pressure and velocity inside

the hole (𝑴∞ = 𝟎.𝟖,∆𝑪𝒑 = 𝟎.𝟎𝟓 )

American Institute of Aeronautics and Astronautics

11

B. Modeling of L/D effect In this section, the L/D effect is estimated from the present computation in Fig. 11. It is the best way that the gradient A in Eq.6 is defined by a function of L/D, however the change of flow rate is nonlinear at high L/D, therefore this way is not appropriate (The nonlinear relation at large L/D is caused by the highly flow rate. The linear relation is shown only when the flow rate is small). Instead of this way, we consider this effect in terms of difference from the flow rate at 𝐿 𝐷⁄ = 1. Figure 14 shows the ratio of flow rate by the value at 𝐿 𝐷⁄ = 1, 𝐹𝐿𝐷. In this figure, changes of 𝐹𝐿𝐷 with the differential pressure are different in each L/D. These changes of 𝐹𝐿𝐷 are approximated by the fitting the results in Fig.14 as following equation

𝐹𝐿𝐷 =(𝜌𝑣)porous

(𝜌𝑣)porous,𝐿 𝐷⁄ =1 (9)

𝐹𝐿𝐷 =

⎩⎪⎨

⎪⎧

0.024 ln(∆𝐶𝑝) + 1.02 (𝐿 𝐷⁄ = 0.5)1 (𝐿 𝐷⁄ = 1)

−0.086 ln(∆𝐶𝑝) + 0.923 (𝐿 𝐷⁄ = 2)−0.271 ln(∆𝐶𝑝) + 0.821 (𝐿 𝐷⁄ = 4)−0.488 ln(∆𝐶𝑝) + 0.666 (𝐿 𝐷⁄ ≥ 8)

� (10)

The flow rate is calculated from the flow rate at 𝐿 𝐷⁄ = 1 and the value of 𝐹𝐿𝐷. The reattachment of flow occurs nearly at L/D=8, then the flow rate does not change even if the 𝐿 𝐷⁄ becomes large. Therefore 𝐹𝐿𝐷 is constant in the range of 𝐿 𝐷⁄ ≥ 8.

VII. Effect of boundary layer In the present study, since we deal with a flow field near wall, the boundary layer condition would be important. In order to investigate how the boundary layer affects the flow rate, the ratio of the boundary layer thickness 𝛿∗ to the hole diameter D, 𝛿∗/𝐷, is changed, and then flow field is computed. The boundary layer thickness 𝛿∗ is the displacement thickness defined as the following equation

𝛿∗ = � �1 −𝑢𝑢∞

�𝑑𝑦∞

0 (11)

Several computation grids with different hole diameters are prepared.

A. Change in flow field by 𝜹∗/𝑫 effect Figure 15 shows the result of flow rate. In both outflow and inflow cases, the change of 𝛿∗/𝐷 affects the flow rate. We discuss the outflow and inflow cases separately as below. a) Outflow cases In the outflow cases, the flow rate increases as 𝛿∗/𝐷 becomes large. To understand this change, Mach number contours and streamlines near the hole edge are shown in Fig. 16 (a). This figure shows the flow passage becomes narrow by the separation and the size of this separation depends on the value of 𝛿∗/𝐷. The schematic of boundary layer effect in outflow cases is shown in Fig. 17 (a). The velocity near the wall (the region surrounded by the red dotted line) is high as 𝛿∗/𝐷 becomes small. The high velocity at the hole edge causes the large separation and it reduces the flow rate through the hole as shown in Fig. 17(a)(II). b) Inflow cases

Figure 15. Variation of mass flow rate (boundary layer effect, 𝑴∞ = 𝟎.𝟖)

American Institute of Aeronautics and Astronautics

12

The flow rate also increases as 𝛿∗/𝐷 is large when the flow goes in. Figure 16 (b) shows Mach number contours and streamlines at the hole edge. In the inflow cases, the flow passage is bent by the wind tunnel flow and the degree of bending depends on the value of 𝛿∗/𝐷. The effect of boundary layer thickness on the flow field of inflow cases is explained in Fig. 17 (b). The bending of flow passage depends on the velocity near the bottom of boundary layer. As 𝛿∗/𝐷 becomes small, the velocity increases in the boundary layer and it causes the large bending. Consequently the flow rate decreases when 𝛿∗/𝐷 is small. The appearance of boundary layer effect is different for outflow or inflow cases. However there is a common characteristic, in which it is important how much the flow of wind tunnel side interact the flow through the hole. This interaction depends on the velocity of boundary layer, and it can be represented as the function of 𝛿∗/𝐷 . Therefore 𝛿∗/𝐷 is an important variable for porous wall model.

B. Modeling of 𝜹∗/𝑫 effect The 𝛿∗/𝐷 effect is extracted from the present computation in Fig. 15. The following equation is created by least-square method

𝐴 = 0.172𝑙𝑛 �𝛿∗

𝐷� + 1.06 (12)

Figure 17. Schematic of flow field near the hole edge

Figure 16. Contours of Mach number near the hole edge (𝑴∞ = 𝟎.𝟖)

American Institute of Aeronautics and Astronautics

13

where the value of A in Eq. 6 is described as a function of 𝛿∗/𝐷.

VIII. Porous wall model by analysis flows through a single hole In Table.2, the present model is summarized.

IX. Comparison of model with direct computation In this section, the present model is applied for a porous wall calculation as a boundary condition and the result

is validated through a comparison with a direct CFD computation in which each hole of the porous wall is simulated. Both computational grids are shown in Fig. 18. The computational methods are same with the previous computations.

Figure 19 shows the comparison of pressure and velocity (vertical component) distributions along the sampling line described as a red dotted line in the upper sketch. The computation result with the present model agrees well with that with direct computation. However some differences are also observed. In the direct computation, the pressure and velocity distributions have small oscillations caused by the flow though each hole as shown in the pressure contours near the porous wall in Fig. 20, whereas no local change appears in the result with the present model. There are also other differences. The flow rate is predicted slightly smaller in the computation with the model, and the variation of pressure at the end of porous wall is somewhat different. However the present model already has enough accuracy for the whole wind tunnel analysis.

Figure 18. Computational grids

Table 2. Present model Present model

Base line (𝜌𝑣)𝑝𝑜𝑟𝑜𝑢𝑠

(𝜌𝑣)∞= 𝐴 ∙ 𝐹𝐿𝐷 ∙ 𝛼 ∙ ∆𝐶𝑝

−0.1 ≤ ∆𝐶𝑝 ≤ 0.1 L/D effect 𝐹𝐿𝐷 =

⎩⎪⎨

⎪⎧

0.024 𝑙𝑛(∆𝐶𝑝) + 1.02 (𝐿 𝐷⁄ = 0.5)1 (𝐿 𝐷⁄ = 1)

−0.086𝑙𝑛(∆𝐶𝑝) + 0.923 (𝐿 𝐷⁄ = 2)−0.271𝑙𝑛(∆𝐶𝑝) + 0.821 (𝐿 𝐷⁄ = 4)−0.488𝑙𝑛(∆𝐶𝑝) + 0.666 (𝐿 𝐷 ≥⁄ 8)

�

Boundary layer effect 𝐴 = 0.172 𝑙𝑛 �𝛿∗

𝐷� + 1.06

American Institute of Aeronautics and Astronautics

14

Figure 20. Contours of pressure distributions

Figure 19. Pressure and velocity distributions above porous wall

American Institute of Aeronautics and Astronautics

15

X. Conclusions For the modeling of porous walls applied in transonic wind tunnels, flows through a hole are directly computed

by a CFD method. We mainly investigate four subjects, differential pressure between wind tunnel side and plenum chamber side (𝑝∞ − 𝑝𝑝𝑙𝑒𝑛𝑢𝑚), free stream Mach number (𝑀∞), ratio of hole’s deepness and diameter (L/D), and boundary layer thickness near the hole edge (𝛿∗/𝐷). From these analyses, we conclude as follows a) Under the small differential pressure such as the condition in JTWT, the flow rate increases linearly with the increase of pressure difference and the flow is largely influenced by the viscous force. Consequently we propose a new model based on a linear relation and it is validated through a comparison with an experiment. This modeling analogizes with the viscous flow theory of Hagen-Poiseuille flow b) The effect of free stream Mach number on the flow rate is treated by nondimensionalizing the flow rate and the differential pressure by the free stream values. This treatment makes it possible to organize one relation between flow rate and differential pressure independent of free stream Mach number. c) The ratio of hole deepness to diameter (L/D) affects the flow field only in the outflow case. The flow rate increases as L/D becomes large. This is because the flow separation area at the hole exit becomes small as the hole becomes deeper. d) The boundary layer thickness affects the flow field in both outflow and inflow cases. When the ratio of boundary layer thickness to hole’s diameter (𝛿∗/𝐷) becomes small, the flow rate decreases. This is because the flow of wind tunnel side interacts the flow through the hole more strongly. e) In addition, the present model is validated by comparing with a direct computation of the flow through a porous wall. The comparison result suggests that the present model has enough accuracy for the simulations of wind tunnel with porous wall.

Acknowledgments The authors would like to thank Prof. Yoshiaki Nakamura, Dr. Koichi Mori, Mr. Hiroto Koyama and Mr.

Kosuke Ishibashi for providing experiment date and useful advices.

References 1 Kohzai, M., Ueno, M., Sudani, N., Hashimoto, A., and Yamamoto, K. “Experimental and Numerical Studies of Support

Interference in the JAXA 2m x 2m Transonic Wind Tunnel”, AIAA paper, 2010-4200. 2 Kohzai, M., Ueno, M., Shiohara, T., and Sudani, N. “Calibration of the test section Mach number in the JAXA 2m x 2m

Transonic Wind Tunnel”, AIAA Paper 2008-848. 3 Ueno, M., and Sudani, N. “Flow Angularity Measurement of 2m x 2m Transonic Wind Tunnel”, AIAA Paper 2008-849. 4 Kohzai, M., Shiohara, T., Ueno, M., Komatsu, Y., Karasawa, T., Koike, A., Sudani, N., Ganaha, Y., Ikeda, M., Watanabe,

A., Haraguchi, T., Nakagawa, M., and Udagawa, D. “Thermal Zero Shift Correction of Strain-Gage Balance Output in the JAXA 2m x 2m Transonic Wind Tunnel”, JAXA-RR-07-034E, 2008.

5 Hudings, M. S., and Hergert, D. W. “Methodology and Results from a Recent Calibration of the Boeing Transonic Wind Tunnel” AIAA Paper 2005-4280.

6 Stojanowski, M., and Germain, E. “The FALCON 7X: from ETW to flight” AIAA Paper 2008-835. 7 Mokry, M. “Two-Dimensional Wind Tunnel Wall Interference” AGARD-AG-281, 1983. 8 Ulbrich, N. “The Application of Panel Method Code ANTARES to Wind Tunnel Wall Interference Problems”, AIAA Paper

2002-307. 9 Hashimoto, A., Aoyama, T., Kohzai, M., and Yamamoto, K. “Transonic Wind Tunnel Simulation with Porous Wall and

Support Devices”, AIAA Paper 2010-4201 10 Slater, J. W. “Improvements in Modeling 90-degree Bleed Holes for Supersonic Inlets” AIAA Paper 2009-710. 11 Abrahamson, K. W., and Bower, D. L. “An Empirical Boundary Condition for Numerical Simulation of Porous Plate

Bleed Flows”, AIAA Paper 1988-306. 12 Harloff, G. J., and Smith, G. E. “On Supersonic-Inlet Boundary-Layer Bleed Flow”, AIAA Paper 1995-38. 13 Akatsuka, J., Watanabe, Y., Murakami, A., and Honami, S. “Porous Bleed model for Boundary Condition of CFD

Analysis”, AIAA paper 2006-3682 14 Frank, M. White. “Viscous Fluid Flow Second edition”, McGraw-Hill, 1991, pp. 429-432. 15 Koyama, H., Ishibashi, K., Mori, K., and Nakamura, Y. “Experiment Investigation of flow through a Hole on a Wind

Tunnel Wall”, 42nd Fluid Dynamics Conference/ Aerospace Numerical Simulation Symposium, Tottori, 2010 (in Japanese)