一定温度麻酔:乱流境界層の流れを研究するツール
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出典:ホセ・ロベルト・モレト、ジェイミー・ドラド、サンディエゴ州立大学航空宇宙工学科、サンディエゴ、カリフォルニア州
境界層は、流れ場に浸漬された固体体の表面に直ちに隣接する薄い流れ領域である。この領域では、粘性せん断応力などの粘性効果が支配し、流体と固体表面との摩擦の影響により流れが遅れる。境界層の外側では、流れがインビシッドされ、すなわち、摩擦、熱伝導または質量拡散による散逸効果がない。
境界層の概念は1904年にルートヴィヒ・プラントルによって導入され、固体上の流れの処理のためのナビエ・ストークス(NS)方程式への大幅な簡素化を可能にした。境界層の内部では、NS方程式は境界層方程式に縮小され、境界層の外側では、NS方程式の簡略化されたバージョンであるオイラー方程式によって流れを記述できます。
図 1.平らな板の上の境界層の開発。
境界層の開発の最も単純なケースは、入射角 0 の平らなプレート上で発生します。平坦なプレート上の境界層の発達を考慮すると、境界層の外側の速度は一定であり、壁に沿った圧力勾配はゼロと見なされます。
固体ボディ表面上で自然に発達する境界層は、通常、次の段階を経る:最初に、層境界層の状態。2 つ目は、遷移状態、3 つ目は乱流境界レイヤーの状態です。各州には、境界レイヤーの流れ構造を記述する独自の法則があります。
境界層の開発と構造に関する研究は、理論的研究と実用化の両方にとって非常に重要です。たとえば、境界層理論は、船舶、航空機、ターボマシンのブレード上の皮膚摩擦ドラッグを計算するための基礎です。皮膚摩擦ドラッグは、境界層内のボディ表面に作成され、それに直接接触する流体粒子を介して表面に加えられる粘性せん断応力によるものです。皮膚摩擦は、流体粘度とサーフェスの局所速度勾配に比例します。皮膚摩擦ドラッグは、表面全体に存在するため、航空機の翼などの大きな領域で重要になります。さらに、乱流流体の流れは、より多くの皮膚摩擦ドラッグを作成します。マクロ乱流流体運動は、高い運動量を持つ流体粒子をサーフェスに持ち込むことにより、境界層内のモメンタム移動を強化します。
このデモでは、混合や加調など、流れが不規則な平坦なプレート上の乱流境界層に焦点を当て、平均流れに変動が重なり合います。したがって、乱流境界層の任意の点の速度は時間の関数です。このデモンストレーションでは、一定温度のホットワイヤ麻酔(CTA)を使用して境界層調査を行います。次に、クラウザー チャート法を使用して、乱流境界層のスキン摩擦係数を計算します。
Princípios
Procedimento
Resultados
The CTA was calibrated in Section 2 of the protocol by measuring the voltage of the hot wire at different air speeds. This data was then used to determine the mathematical relationship between the measured variable, voltage, and the indirect variable, air speed. There are many approaches to fitting the experimental data to mathematical relationships for velocity, several of which are covered in the appendix. After the mathematical relationship is determined, velocity is easily calculated from the voltage in further experiments with the CTA.
In section 3 of the protocol, the air speed was measured using the CTA at different vertical positions in the wind tunnel. This represented different distances, y, from the flat plate. From the measured instantaneous flow velocity at each point, the average boundary layer velocity profile can be obtained. The velocity profile, u(y), can be used to determine the vertical distance that the plate would have to be moved perpendicular to itself for an inviscid flow to obtain the same flow rate that occurs between the surface and fluid, called the boundary layer displacement thickness, 
*. This is defined as:
where 
is the free stream velocity. The momentum thickness, θ, or the distance the plate would have to be moved in the direction parallel to itself in order to have the same momentum that exists in between the fluid and itself, is defined as:
Then, the shape factor, H, which can be used to determine the nature of the flow, is defined as:
where a shape factor of 1.3 indicates fully turbulent flow, a shape factor of 2.6 indicates laminar flow, and any value in between represents transition or turbulent yet not fully developed flow.
For the turbulent boundary layer case, several properties can be further examined. The skin friction can be determined using the Clauser chart method (see Figure 4). The Clauser chart method can be used to obtain the skin friction coefficient, Cf, from the measured velocity, u(y). From log law-of-the-wall, we have the following:
where κ ≈ 0.40 ~ 0.41 and B=5.0 to 5.5. Practically, κ=0.4 and B=5.5. From the definition, the skin friction coefficient is given by:
where q is the dynamic pressure of the free stream and τw is the shear stress at the wall. The log law-of-the-wall can then be expressed as (See Appendix):
where, 
.
Given a series of Cf values, a family of curves can be generated for 
vs. Ry. Several values of Ry ranging from 100 to 100,000 and Cf values ranging from 0.001 to 0.006 should be used to plot the curves in a log-linear format. This forms the Clauser chart, which can be used to determine the skin friction coefficient, Cf, as shown in Figure 4. By comparing the measured boundary layer velocity profile with the family of curves that are based on the log law-of-the-wall with the prescribed skin-friction coefficient values, the curve that best overlaps with the log law portion of the measured velocity profile gives the value of the measured skin friction coefficient.
Figure 4: Clauser Chart.
This result can be compared to the result obtained using the integral equation method. Also, the velocity fluctuation profile can be obtained and the experimental result can be compared against the log law-of-the-wall. See the Appendix for more information.
Applications and Summary
The demonstration shows how to use constant temperature anemometry, a powerful tool used to study turbulent flow over a surface, which in this specific case was a flat plate. This method is simpler and less expensive than other methods, such as PIV, PTV, and LDV, and it provides a high temporal resolution. The application of hot-wire anemometry to a turbulent boundary layer provides a cost effective and hands-on approach to demonstrate the behavior of turbulent flows.
Constant temperature anemometry has numerous applications. This technique can be used to survey both turbulent and laminar flows. Hot-wire anemometry can be used to study the wake flows of an airfoil or an airplane model, thus providing information such as the drag of the airfoil and the level of wake turbulence, which provides valuable information for aircraft design.
Hot-wire anemometry can also be used in environmental fluid dynamics investigations, such as to study plume flows, which are responsible for the mass and momentum transport and mixing of a variety of processes found in the Earth`s atmosphere.
A variation to hot-wire anemometry is hot-film anemometry, which is typically used in liquid flows that require robust and reliable performance. For example, monitoring of the air flow at the air intake duct of an automobile engine is often performed by a sensor made of hot film.
The application of hotwire anemometry is not restricted to the mechanical engineering realm. CTA can also be used for example in biomedical applications to measure respiration rate.
Materials List
| Name | Company | Catalog Number | Comments |
| Equipment | |||
| Instructional Subsonic Wind Tunnel | Jetstream | The dimensions of the test section of the wind tunnel are as follows: 5.25" (width) x 5.25"(height) x 16" (length). The wind tunnel should be able to attain air speeds of 0 – 80 mph. | |
| The Wall | The wall of the test section is made of glass. | ||
| CTA model 1750 | TSI Corp. | ||
| Hot-wire probe | TSI Corp | TSI 1218-T1.5 | Tungsten-platinum coated, standard boundary layer probe. The diameter of the probe is 3.81 μm. The length of the sensing area of the wire is 1.27 mm. |
| A/D Board | National Instruments | NI USB 6003 | Maximum sampling rate of 100 kHz with 16-bit resolution |
| Traverse System | Newport | Newport 370-RC Rack-And-Pinion Rod Clamp & 75 Damped Optical Support Rod Assembly | |
| Pitot tube | The dynamic pressure of the free stream will be sensed by a tiny Pitot tube installed at the beginning region of the test section. The resolution of the Pitot tube is 0.1 mph. | ||
| Software | LabView software will be used for data acquisition. | ||
| Power Supply | Heath | 2718 | Heath 2718 Tri-Power Supply with 15V DC output is used to power the hot-wire anemometer. |
| Oscilloscope | Tektronix | 2232 | |
| Signal Generator | Agilent | 33110A |
Transcrição
A boundary layer is a thin flow region immediately adjacent to the surface of a solid body in a flow field. The region of flow outside of the boundary layer, called the free stream region has a constant velocity. However, within the boundary layer there is a velocity gradient due to friction at the surface. The boundary layer typically undergoes several stages.
First the laminar boundary state, followed by the transition state and finally, the turbulent boundary layer state, which involves irregular flow and fluctuations, like mixing or eddying. The boundary layer is the basis for the calculation of skin friction drag on aircraft.
Skin friction drag is created within the boundary layer and is due to the viscous shear stress exerted on the surface. Skin friction drag is proportional to fluid dynamic viscosity, mu, and the local stream wise velocity shear strain rate, which is the gradient of the streamwise velocity in the normal direction. So it becomes significant for large areas, such as an airplane wing. Additionally, skin friction drag is higher in turbulent flow, since the fluid particles interact with the surface at high momentum.
One way to measure turbulent boundary layer properties is using hot wire anemometry, which is based on two principles related to the cooling effect of flow on a heated wire. According to the first principle, when a fluid flows over a hot surface, the convective heat coefficient changes, which results in changes in the surface temperature.
The second principle is Joule’s law, which states that an electrical conductors heat dissipation, Q, is proportional to the square of the electric current, I, applied to the conductor. We can use the two principles to determine the velocity of fluid flow surrounding a heated metallic wire probe, by measuring the electrical potential E, that has to be applied to maintain a constant temperature of the wire.
A commonly used hot wire technique is Constant Temperature Anemometry or CTA. CTA consists of a very thin metallic wire, called the probe, which is connected to the arm of a Wheatstone bridge. The Wheatstone bridge controls the electrical potential and adjusts it as needed in order to maintain a constant temperature across the wire. Any cooling is caused by fluid flow around the wire. Thus, the change in the potential is a function of the heat transfer coefficient and by extension is a function of velocity.
In this experiment, we will demonstrate the use of a Constant Temperature Anemometry setup to measure the turbulent boundary layer over a flat plate.
First, we will learn how the Constant Temperature Anemometer, or CTA, system responds to flow signal changes using a wind tunnel. To begin, secure the hot wire probe of the CTA system inside of the wind tunnel using a support shaft.
Then, set up a DC power supply, signal generator, and oscilloscope. The components are connected as shown. To begin, turn on the hot wire power supply, the signal generator and the oscilloscope. Set the signal generator to supply a square wave input to the Wheatstone bridge with a 150 mV amplitude and a 10 kHz frequency.
Observe the output signal in the oscilloscope to make sure that the frequency and amplitude are correct. Now close the test section, plug in the serial cable, turn on the wind tunnel and set the wind speed to 40 mph. Once the airflow stabilizes, measure the width of the signal overshoot, tau, observed on the oscilloscope. Use the measured value of tau to calculate the cut-off frequency for the hot wire system using this equation. Then turn off the wind tunnel airflow.
Next we will establish the correlation between wind speed and the electrical potential of the Wheatstone bridge. To begin, raise the CTA probe in the vertical direction so that it is in the free stream region. Start the wind tunnel control software and then start the virtual instrument software. Set the sampling rate to 10 kHz and the number of samples to 100,000.
Now, with the wind tunnel airspeed set to 0 mph, record the voltage on the Wheatstone bridge. Then, increase the wind tunnel airspeed at increments of 3 mph up to 15 mph, measuring the voltage at each increment. Be sure to allow the air flow to stabilize before recording the voltage measurement.
Next, increase the wind tunnel air speed up to 60 mph in 5-mph increments, measuring the voltage at each increment. When all measurements are complete, reduce the airspeed to 30 mph and then turn off the wind tunnel airflow.
Using the same setup as before, lower the CTA probe slowly until it touches the test section floor, which will act as the flat plate. Set the airflow to 40 mph. Keep the sampling frequency at 10 kilohertz and the number of samples at 100,000. Record the voltage at the lowest vertical setting, which is next to the flat plate and in the boundary layer.
Now, move the probe vertically in steps of 0.05 mm up to a height of 0. 5 mm, recording the voltage reading at each position. Then, increase the probe height in increments of 0.1 mm up to a height of 1. 5 mm. And then in steps of 0.25 mm up to a final height of 4 mm, while recording the voltage at each increment.
When all of the measurements have been made, reduce the wind speed to 20 mph and then turn off the airflow. Then shut off the power supply, signal generator, and oscilloscope.
The first step in analyzing the data is to use the data acquired during the calibration step of the experiment, to determine the correlation between the hot wire voltage and air speed. There are several different methods to do this, which involve fitting the data to known heat transfer relationships, and it’s covered in detail in the appendix of this video.
Once the mathematical relationship has been determined, use the voltage measurements to calculate velocity at each vertical height. After adjusting the nominal height to account for any artifacts from an overbent probe, plot the velocity profile u(y), which can then be used to determine the boundary layer displacement thickness.
This value represents the distance that the plate would have to be moved vertically in order to obtain the same flow rate that occurs between the surface and the fluid. We can also calculate the momentum thickness, defined as shown, which is the distance the plate would have to be moved vertically in order to have the same momentum that exists between the fluid and plate.
From these two parameters, we can calculate the shape factor, H. The shape factor is used to determine the nature of the flow, where a shape factor of about 1.3 indicates fully turbulent flow and about 2.6 for laminar flow. Between these values is transitional flow. In the case of this experiment, the shape factor was calculated as 1.9, indicating transitional flow.
In summary, we learned about boundary layer flow development, and then used a Constant Temperature Anemometry setup to analyze the turbulent boundary layer over a flat plate and observe low behavior.