Numerical Model for Five-Hole Probe Response

The analytical approach described in the previous section is not ideal for the current research due to its limited capabilities to adequately model the behavior of a 5HP. While there are likely methods of solving the problems discussed above this section will introduce an alternative to Eq. 5.6. The method discussed below can be used to numerically simulate the measurements of a five-hole probe in an arbitrary turbulent flowfield. In Chapter 6 the ability to predict the response of a standard five-hole probe to an arbitrary turbulent flow will be used to understand how turbulence effects the five-hole probe measurements and to develop a method of correcting these measurements.

The instantaneous pressure at a pressure port on a five-hole probe was expressed using Eq. 5.4. Since the goal is to determine the influence of turbulent velocity fluctuations on the measured port pressures it will be convenient to rewrite the instantaneous port pressure coefficient in terms of the velocity components rather than the flow angles.

𝐶

_𝑝,𝑖

(𝑡) = 𝐶

_𝑝𝑠

(𝑡) + 𝑉

𝑡2

𝑈

_∞2

(𝑡) 𝑓

𝑖

( 𝑈(𝑡) , 𝑉(𝑡) , 𝑊(𝑡) )

Eq. 5.11

Time averaging this expression results in the following equation for the average pressure coefficient measured by the individual ports.

113

𝐶

_𝑝,𝑖

̅̅̅̅̅ = 𝐶̅̅̅̅ +

_𝑝𝑠

1

𝑈

_∞2

𝑉̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅

𝑡2

(𝑡) 𝑓

𝑖

( 𝑈(𝑡) , 𝑉(𝑡) , 𝑊(𝑡) )

Eq. 5.12

This equation represents a more general form of Eq. 5.6. Equation 5.6 was derived by assuming a functional form of the 2nd term on the right hand side of Eq. 5.12. The numerical simulation introduced in this section will instead evaluate this term numerically.

While the Navier Stokes equations represent a deterministic set of equations it is often a valid assumption to treat the turbulent velocity fluctuations as a random process. With this assumption the instantaneous velocity and the port angularity function 𝑓_𝑖 can also be considered random functions of time. Assume that 𝑔(𝑥) is an arbitrary but known function of 𝑥 where 𝑥 is a random variable with a probability distribution function (p.d.f) 𝒫𝑥. The mean value of 𝑔(𝑥) can be determined by integrating 𝑔(𝑥) weighted by the probability of 𝑥 over all possible values of 𝑥.65

𝑔̅ = ∫ 𝑔(𝑥) 𝒫

∞ _𝑥

𝑑𝑥

−∞

Eq. 5.13 This process can be used to determine the value of the 2nd term on the right hand side of Eq. 5.12. First, define 𝒫_𝑈𝑉𝑊 as the joint probability distribution function of the three instantaneous velocity components ( 𝑈(𝑡) , 𝑉(𝑡) , 𝑊(𝑡) ) in a turbulent flow. The mean value of the quantity 𝑉_𝑡2 _𝑓

𝑖(𝑈 , 𝑉 , 𝑊) can be expressed as:

𝑉

_𝑡2

_𝑓

𝑖

(𝑈 , 𝑉 , 𝑊)

̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ = ∭[(𝑈

2

_{+ 𝑉}

2

_{+ 𝑊}

2

_{) 𝑓}

𝑖

(𝑈, 𝑉, 𝑊)] 𝒫

𝑢𝑣𝑤

𝑑𝑈𝑑𝑉𝑑𝑊

∞ −∞ Eq. 5.14 Note that the explicit dependence of velocity on time (𝑡) has been dropped for clarity, and the definition of the instantaneous total velocity 𝑉_𝑡, see Eq. 2.2, was used on right hand side of Eq. 5.14. It is also assumed that the turbulence is statistically stationary so that the probability distribution function does not depend on time. In order to evaluate the triple integral in Eq. 5.14

114

the port angularity function 𝑓_𝑖and the joint probability distribution function of the velocity components 𝒫𝑈𝑉𝑊 must be known. The probability distribution function can be approximated using a multivariate normal distribution function.66

𝒫

_𝑢𝑣𝑤

=

1

√(2𝜋)

3

_|Σ

𝑢𝑣𝑤

|

𝑒𝑥𝑝 (

−1₂

[𝑋

_𝑢𝑣𝑤𝑇

_Σ

𝑢𝑣𝑤−1

𝑋

𝑢𝑣𝑤

])

Eq. 5.15 Where 𝑋𝑢𝑣𝑤 is a 1-by-3 matrix:

𝑋

_𝑢𝑣𝑤

= [

𝑈(𝑡) − 𝑈̅

𝑉(𝑡) − 𝑉

𝑊(𝑡) − 𝑊̅

]

𝑇 Eq. 5.16

Note that the superscript T denotes the transpose operation. Σ_𝑢𝑣𝑤 is the 3-by-3 covariance matrix also known as the Reynolds Stress Tensor.56,66

Σ

_𝑢𝑣𝑤

= [

𝑢′

2

̅̅̅̅

_{𝑢′𝑣′̅̅̅̅̅ 𝑢′𝑤′}̅̅̅̅̅̅

𝑢′𝑣′

̅̅̅̅̅

_𝑣′̅̅̅̅

2

_{𝑣′𝑤′}̅̅̅̅̅̅

𝑢′𝑤′

̅̅̅̅̅̅ 𝑣′𝑤′̅̅̅̅̅̅

_𝑤′̅̅̅̅̅

2

]

Eq. 5.17

The three diagonal terms (𝑢′̅̅̅̅, 𝑣′2 ̅̅̅̅, 𝑤′2 ̅̅̅̅̅) are known as the Reynolds normal stresses while the 2 off-diagonal terms (𝑢′𝑣′̅̅̅̅̅, 𝑢′𝑤′̅̅̅̅̅̅, 𝑣′𝑤′̅̅̅̅̅̅) are known as the Reynolds shear stresses. It should be noted that strictly speaking turbulence is not adequately represented by a Gaussian probability distribution due to the behavior of higher-order statistics.56,66

Now that the probability distribution function is defined the next step is to evaluate the port angularity function 𝑓𝑖(𝑈 , 𝑉 , 𝑊). At a given instant in time, if the velocity components are known the value of the port angularity function can be determined using the experimental probe calibration data. The most straightforward way to accomplish this is to calculate the pitch and yaw angles (𝛼, 𝛽) from the velocity components, see Eqs. 4.17 to 4.19, and then interpolate within the calibration dataset, see Eq. 5.2.

115

Now with the use of Eq. 5.2 and Eqs. 5.14 through 5.17 the measurements of a 5HP in an arbitrary flowfield can be simulated with the following procedure.

1. Select values for the mean static pressure (𝑃̅ ), the mean velocity components (𝑈̅, 𝑉̅, 𝑊𝑠 ̅ ) and the 6 components of the Reynolds Stress Tensor, see Eq. 5.17.

2. Artificial time histories of the velocity components (𝑈(𝑡), 𝑉(𝑡), 𝑊(𝑡)) can now be created by generating random values from the multivariate normal distribution Eq. 5.15. 3. For each “instant in time” calculate the instantaneous pitch and yaw angles (𝛼(𝑡), 𝛽(𝑡))

from the velocity components using Eqs. 4.17 to 4.19.

4. For each pressure port, determine the instantaneous value of the port angularity function 𝑓_𝑖(𝛼(𝑡), 𝛽(𝑡)) by interpolating within the calibration dataset using Eq. 5.2.

5. By generating a sufficiently long artificial time history an accurate mean of the quantity 𝑉_𝑡2 _𝑓

𝑖(𝑈 , 𝑉 , 𝑊) can be determined and the average port pressures (𝑃̅) can be calculated 𝑖 using Eq. 5.12.

6. Once the five average port pressures (𝐶̅̅̅̅) are known the “simulated measurements” can _𝑝,𝑖 be processed using the probe calibration data and the procedure described in Section 4.2. This procedure will be validated in Chapter 6 and it will be used to study and model the effects of turbulence on standard five-hole probes. Then it will be demonstrated that this simulation method can be used to develop a correction process for five-hole probes.

It is worth pointing out that the process described above is similar to that used by Becker and Brown15 in their derivation of a model for the response of a Pitot probe to turbulent flow, see Eq. 2.12. They arrived at an expression similar to Eq. 5.14 but they assumed a specific functional form of the general angularity function 𝑓_𝑖, see Eq. 2.10. They then made several simplifying assumptions in order to arrive at Eq. 2.12. For example, they assumed the probe was aligned with

116

the mean flow (𝑉̅ = 𝑊̅ = 0), the velocity fluctuations were uncorrelated and they imposed certain relationships on the statistics of the velocity fluctuations such as 𝑣′̅̅̅̅ = 𝑤′2 ̅̅̅̅̅ and 𝑢′2 ̅̅̅̅ =2 𝑐(𝑣′̅̅̅̅ + 𝑤′2 ̅̅̅̅̅). In isotropic turbulence 𝑐 = 1 and in turbulent shear flow 𝑐 = 1/2. In addition, to 2 analytically evaluate the triple integral in Eq. 5.14 they took advantage of the axisymmetric geometry of the Pitot probe which cannot be done for the outer ports on a five-hole probe. The numerical simulation described above is a more general approach than that of Becker and Brown15 with several important advantages. First, the numerical simulation places no restrictions on the functional form of the port angularity function which is especially important for the outer ports of the 5HP since symmetry cannot be used. In addition, the functional form assumed by Becker and Brown cannot be expected to remain valid for an arbitrary range of angles, but when directly using the probe calibration data the maximum angular range is only limited by the maximum angle in the dataset. Second, the probe does not have to be aligned with the mean flow, and finally no restrictions are placed on the turbulent velocity fluctuations other than the assumptions that their probability distribution function can be described by a multivariate normal distribution.

It should be mentioned that this simulation technique does not account for all possible factors that influence five-hole probe measurements. Recall that the quasi-steady assumption states that if the length scale of the turbulent fluctuation is large compared to the probe tip dimensions then the instantaneous flow over the tip is equivalent to a steady laminar flow.15 Turbulent flows are very complex with energy distributed over a wide range of length scales.56,66 Except perhaps at very low Reynolds numbers the turbulent flow will almost certainly contain energy at scales smaller than the probe tip. How this may influence a multi-hole probe is unknown and is not investigated in this research. The models described above implicitly assume

117

that the 5HP measurements are true point measurements, but it is known that velocity gradients across the tip of the probe can affect the measurements of a 5HP. Ligrani et al.11 developed an empirical correction scheme for 5HPs in a velocity gradient. Their empirical method could be included in the simulation method described above but the effect of the velocity gradient is dependent on probe geometry and their model requires calibration of the probe in a velocity gradient. An additional effect that is not being included is the influence of the instantaneous acceleration known as the apparent-mass effect.43,63 Johansen and Rediniotis63 investigated the influence of acceleration on a fast-response five-hole probe and concluded that this effect is typically very small.

In document The effects of turbulence on the measurements of five-hole probes (Page 122-127)