**3.2. Contraction**

Taking into account that it is sometimes necessary to place additional equipment, e.g. meas‐ uring instruments, supports, etc., inside the test chamber, it is convenient to maintain the operation pressure inside it equal to the local environment pressure. To fulfil this condition, it is recommended to have a small opening, approximately 1,0% of the total length of the test

From the point of view of the pressure loss calculation, the test chamber will be considered as a constant section duct with standard finishing surfaces. Nevertheless, in some cases, the test chamber may have slightly divergent walls, in order to compensate for the boundary layer growth. This modification may avoid the need for tail flotation correction for aircraft model

Figure 2 shows a design of a typical constant section test chamber. With the typical dimensions and velocities inside a wind tunnel, the flow in the test section, including the boundary layer, will be turbulent, because it is continuous along the whole wind tunnel. According to Idel´Cik (1969), the pressure loss coefficient, related to the dynamic pressure in the test section, which is considered as the reference dynamic pressure for all the calculations, is given by the

tests, although it would be strictly valid only for the design Reynolds number.

chamber, at the entrance of the diffuser 1.

8 Wind Tunnel Designs and Their Diverse Engineering Applications

**Figure 2.** Layout of a constant section wind tunnel test chamber.

expression:

*ζ* =*λ* · *L* / *DH* ,

The contraction or "nozzle" is the most critical part in the design of a wind tunnel; it has the highest impact on the test chamber flow quality. Its aim is to accelerate the flow from the settling chamber to the test chamber, further reducing flow turbulence and non-uniformities in the test chamber. The flow acceleration and non-uniformity attenuations mainly depend on the so-called contraction ratio, *N*, between the entrance and exit section areas. Figure 3 shows a typical wind tunnel contraction.

**Figure 3.** General layout of a three-dimensional wind tunnel contraction.

Although, due to the flow quality improvement, the contraction ratio, *N*, should be as large as possible, this parameter strongly influences the overall wind tunnel dimensions. Therefore, depending on the expected applications, a compromise for this parameter should be reached.

Quoting P. Bradshaw and R. Metha (1979), "The effect of a contraction on unsteady velocity variations and turbulence is more complicated: the reduction of x-component (axial) fluctua‐ tions is greater than that of transverse fluctuations. A simple analysis due to Prandtl predicts that the ratio of root-mean-square (rms) axial velocity fluctuation to mean velocity will be reduced by a factor 1/*N*<sup>2</sup> , as for mean-velocity variations, while the ratio of lateral rms fluctuations to mean velocity is reduced only by a factor of *N*: that is, the lateral fluctuations (in m/s, say) increase through the contraction, because of the stretching and spin-up of elementary longitudinal vortex lines. Batchelor, *The Theory of Homogeneous Turbulence*, Cambridge (1953), gives a more refined analysis, but Prandtl's results are good enough for tunnel design. The implication is that tunnel free-stream turbulence is far from isotropic. The axial-component fluctuation is easiest to measure, e.g. with a hot-wire anemometer, and is the "free-stream turbulence" value usually quoted. However, it is smaller than the others, even if it does contain a contribution from low-frequency unsteadiness of the tunnel flow as well as true turbulence."

In the case of wind tunnels for civil or industrial applications, a contractions ratio between 4,0 and 6,0 may be sufficient. With a good design of the shape, the flow turbulence and nonuniformities levels can reach the order of 2,0%, which is acceptable for many applications. Nevertheless, with one screen placed in the settling chamber those levels can be reduced up to 0,5%, which is a very reasonable value even for some aeronautical purposes.

For more demanding aeronautical, when the flow quality must be better than 0,1% in nonuniformities of the average speed and longitudinal turbulence level, and better than 0,3% in vertical and lateral turbulence level, a contraction ratio between 8,0 and 9,0 is more desirable. This ratio also allows installing 2 or 3 screens in the settling chamber to ensure the target flow quality without high pressure losses through them.

The shape of the contraction is the second characteristic to be defined. Taking into account that the contraction is rather smooth, one may think that a one-dimensional approach to the flow analysis would be adequate to determine the pressure gradient along it. Although this is right for the average values, the pressure distribution on the contraction walls has some regions with adverse pressure gradient, which may produce local boundary layer separation. When it happens, the turbulence level increases drastically, resulting in poor flow quality in the test chamber.

According to P. Bradshaw and R. Metha (1979), "The old-style contraction shape with a small radius of curvature at the wide end and a large radius at the narrow end to provide a gentle entry to the test section is not the optimum. There is a danger of boundary-layer separation at the wide end, or perturbation of the flow through the last screen. Good practice is to make the ratio of the radius of curvature to the flow width about the same at each end. However, a too large radius of curvature at the upstream end leads to slow acceleration and therefore increased rate of growth of boundary-layer thickness, so the boundary layer - if laminar as it should be in a small tunnel - may suffer from Taylor-Goertler "centrifugal'' instability when the radius of curvature decreases".

According to our experience, when both of the contraction semi-angles, *α*/2 and *β*/2 (see Figure 3), take the values in the order of 12º, the contraction has a reasonable length and a good fluid dynamic behaviour. With regard to the contour shape, following the recommendations of P. Bradshaw and R. Metha (1979), two segments of third degree polynomial curves are recom‐ mended.

**Figure 4.** Fitting polynomials for contraction shape.

Quoting P. Bradshaw and R. Metha (1979), "The effect of a contraction on unsteady velocity variations and turbulence is more complicated: the reduction of x-component (axial) fluctua‐ tions is greater than that of transverse fluctuations. A simple analysis due to Prandtl predicts that the ratio of root-mean-square (rms) axial velocity fluctuation to mean velocity will be

fluctuations to mean velocity is reduced only by a factor of *N*: that is, the lateral fluctuations (in m/s, say) increase through the contraction, because of the stretching and spin-up of elementary longitudinal vortex lines. Batchelor, *The Theory of Homogeneous Turbulence*, Cambridge (1953), gives a more refined analysis, but Prandtl's results are good enough for tunnel design. The implication is that tunnel free-stream turbulence is far from isotropic. The axial-component fluctuation is easiest to measure, e.g. with a hot-wire anemometer, and is the "free-stream turbulence" value usually quoted. However, it is smaller than the others, even if it does contain a contribution from low-frequency unsteadiness of the tunnel flow as well as

In the case of wind tunnels for civil or industrial applications, a contractions ratio between 4,0 and 6,0 may be sufficient. With a good design of the shape, the flow turbulence and nonuniformities levels can reach the order of 2,0%, which is acceptable for many applications. Nevertheless, with one screen placed in the settling chamber those levels can be reduced up

For more demanding aeronautical, when the flow quality must be better than 0,1% in nonuniformities of the average speed and longitudinal turbulence level, and better than 0,3% in vertical and lateral turbulence level, a contraction ratio between 8,0 and 9,0 is more desirable. This ratio also allows installing 2 or 3 screens in the settling chamber to ensure the target flow

The shape of the contraction is the second characteristic to be defined. Taking into account that the contraction is rather smooth, one may think that a one-dimensional approach to the flow analysis would be adequate to determine the pressure gradient along it. Although this is right for the average values, the pressure distribution on the contraction walls has some regions with adverse pressure gradient, which may produce local boundary layer separation. When it happens, the turbulence level increases drastically, resulting in poor flow quality in the test

According to P. Bradshaw and R. Metha (1979), "The old-style contraction shape with a small radius of curvature at the wide end and a large radius at the narrow end to provide a gentle entry to the test section is not the optimum. There is a danger of boundary-layer separation at the wide end, or perturbation of the flow through the last screen. Good practice is to make the ratio of the radius of curvature to the flow width about the same at each end. However, a too large radius of curvature at the upstream end leads to slow acceleration and therefore increased rate of growth of boundary-layer thickness, so the boundary layer - if laminar as it should be in a small tunnel - may suffer from Taylor-Goertler "centrifugal'' instability when the radius

to 0,5%, which is a very reasonable value even for some aeronautical purposes.

quality without high pressure losses through them.

, as for mean-velocity variations, while the ratio of lateral rms

reduced by a factor 1/*N*<sup>2</sup>

10 Wind Tunnel Designs and Their Diverse Engineering Applications

true turbulence."

chamber.

of curvature decreases".

As indicated in Figure 4, the conditions required to define the polynomial starting at the wide end are: the coordinates (*xW,yW*), the horizontal tangential condition in that point, the point where the contour line crosses the connection strait line, usually in the 50% of such line, and the tangency with the line coming from the narrow end. For the line starting at the narrow end the initial point is (*xN,yN*), with the same horizontal tangential condition in this point, and the connection to the wide end line. Consequently, the polynomials are:

$$y = a\_W + b\_W \cdot \mathbf{x} + c\_W \cdot \mathbf{x}^2 + d\_W \cdot \mathbf{x}^3,$$

$$y = a\_N + b\_N \cdot \mathbf{x} + c\_N \cdot \mathbf{x}^2 + d\_N \cdot \mathbf{x}^3.$$

Imposing the condition that the connection point is in the 50%, the coordinates of that point are [*xM*,*yM*]=[(*xW*+*xN*)/2,(*yW*+*yN*)/2)]. Introducing the conditions in both polynomial equations, the two families of coefficients can be found.

According to Idel´Cik (1969), the pressure loss coefficient related to the dynamic pressure in the narrow section, is given by the expression:

$$\mathcal{L} = \left\{ \frac{\lambda}{\prod\_{16} \cdot \sin\left(\frac{\alpha}{2}\right)} \right\} \left(1 - \frac{1}{N^2}\right) + \left\{ \frac{\lambda}{\prod\_{16} \cdot \sin\left(\frac{\beta}{2}\right)} \right\} \left(1 - \frac{1}{N^2}\right),$$

where *λ* is defined as:

$$
\lambda = 1 / \text{(1,8 log Re - 1,64)}^2.
$$

The Reynolds number is based on the hydraulic diameter of the narrow section.
