Dynamic Similarity

J. Gordon Leishman

doi:https://doi.org/10.15394/eaglepub.2022.1066.n14

19 Dynamic Similarity

Introduction

The principle of dynamic similarity is closely related to the concepts of dimensional analysis. The need for similarity or similitude arises whenever subscale engineering models are tested in the laboratory or in a wind tunnel. A subscale model application is considered similar to the actual (real) application if the two share the same geometric, kinematic, and dynamic characteristics. In general, geometric similarity focuses on the scaling of dimensions, kinematic similarity on the scaling of displacements and velocities, and dynamic similarity on the scaling of integrated values such as forces. Prerequisites for dynamic similarity are geometric and kinematic similarity. Therefore, any claim that “dynamic similarity” has been achieved implies that geometric and kinematic similarity conditions have also been met.

Dynamic similarity can be achieved by matching the values of the similarity parameters across different circumstances, such as in two or more separate experiments or at the model and full-scale levels. However, achieving this goal in practice can be challenging. If the similarity parameters are equal, the physics of both situations will be correctly scaled, ensuring they exhibit the same physical similarity. As previously discussed, the critical aerodynamic similarity parameters are the Reynolds and Mach numbers. However, in general, many other relevant similarity parameters arise in aerodynamics and engineering problem-solving, including the Froude, Weber, Strouhal, Stokes, and Prandtl numbers. Similarity parameters of various types are used in all fields of engineering. Therefore, understanding and applying the principles of dynamic similarity is essential in engineering analyses.

Learning Objectives

Understand the concepts of geometric, kinematic, and dynamic similarity, and appreciate their significance as tools for engineering problem-solving.
Use dimensional analysis to determine how problem parameters scale to achieve dynamic similarity, i.e., how the problem scales with geometric size, operating conditions, and related parameters.
Appreciate the challenges of using scaled models in the laboratory and wind tunnel testing to obtain measurements that can be applied to full-scale flight vehicles.
Realize the benefits and limitations of testing actual (full-scale) problems on a smaller scale and why only partial similarity may be possible in some cases.

Similarity Criteria

The following formal criteria are required to achieve flow similarity or similitude between the model or sub-scale test article and the eventual or actual application:

Geometric similarity means the sub-scale model is correctly scaled and has the same geometric shape as the actual application. Thus, a single scaling parameter can relate the two geometries.
Kinematic similarity means that consistent scale factors must relate corresponding displacements and velocities in the sub-scale model and the actual application.
Dynamic similarity means that the ratios of all forces and moments in the sub-scale model and the actual application are the same, i.e., inertial, gravitational, viscous, pressure, elastic, and surface forces and moments. If dynamic similarity is achieved, the geometric and kinematic similitude conditions will also be satisfied.

Suppose all relevant similarity parameters for a given problem can be matched. In this case, it can be ensured that the physics of both situations is correctly scaled and has the correct physical similarity. Therefore, engineers can typically study the physical characteristics of a full- or larger-scale system (e.g., one still in the design process) at a smaller scale under controlled laboratory or wind-tunnel conditions. Unfortunately, matching all relevant similarity parameters at a smaller scale is difficult, so only partial similarity may be possible. In that case, there are still techniques that can be applied to achieve the desired results for the full-scale problem.

Navier-Stokes Equations

The significance of the Reynolds and Mach numbers can also be determined from the Navier-Stokes equations of fluid motion. The incompressible form of the Navier–Stokes equation can be written as

(1) $\begin{equation*} \varrho \frac{D\vec{V}}{Dt} = -\nabla p + \mu \, \nabla^2 \vec{V} \end{equation*}$

which is a vector equation in which body forces, such as gravity, have been omitted. The Navier-Stokes equations are among the most fundamental governing equations of fluid dynamics. Dividing Eq. 1 through by $\varrho$ gives

(2) $\begin{equation*} \frac{D\vec{V}}{Dt} = -\frac{1}{\varrho}\nabla p + \frac{\mu}{\varrho} \, \nabla^2 \vec{V} \end{equation*}$

If velocity, position, time, and pressure are scaled using the reference quantities $V_\infty$ , $L$ , and $\varrho_\infty V_\infty^2$ , then the nondimensional variables are

(3) $\begin{equation*} \vec{U}=\frac{\vec{V}}{V_\infty}, \qquad \vec{x}^{\,*}=\frac{\vec{x}}{L}, \qquad t^*=\frac{tV_\infty}{L}, \qquad p^*=\frac{p}{\varrho_\infty V_\infty^2} \end{equation*}$

Using these variables, the nondimensional incompressible Navier-Stokes equation becomes

(4) $\begin{equation*} \frac{D\vec{U}}{Dt^*} = -\nabla^* p^* + \frac{1}{Re}\nabla^{*2}\vec{U} \end{equation*}$

where

(5) $\begin{equation*} Re = \frac{\varrho_\infty V_\infty L}{\mu} \end{equation*}$

For compressible flows, the Mach number also appears, i.e.,

(6) $\begin{equation*} M = \frac{V_\infty}{a_\infty} \end{equation*}$

If pressure is scaled instead by the thermodynamic reference pressure $p_\infty$ , define

(7) $\begin{equation*} \widehat{p}=\frac{p}{p_\infty} \end{equation*}$

Then the nondimensional pressure-gradient term takes the form

(8) $\begin{equation*} -\frac{1}{\gamma M^2}\nabla^* \widehat{p} \end{equation*}$

because $a_\infty^2=\gamma p_\infty/\varrho_\infty$ . Therefore, the nondimensional momentum equation may be written schematically as

(9) $\begin{equation*} \frac{D\vec{U}}{Dt^*} = -\frac{1}{\gamma M^2}\nabla^* \widehat{p} + \frac{1}{Re}\nabla^{*2}\vec{U} \end{equation*}$

This nondimensionalization of the governing momentum equation reveals two essential properties:

Dynamic Similarity: Flows about geometrically similar shapes that share the same $Re$ , $M$ , and any other relevant similarity parameters, together with similar boundary and initial conditions, will exhibit dynamically similar behavior regardless of the absolute scales of ${V_{\infty}}$ and $L$ .
Flow Regime Effects: The Reynolds number quantifies the ratio of inertial to viscous forces. A higher $Re$ indicates that inertial effects are more important, reducing the relative influence of viscosity. The Mach number is a measure of compressibility effects. At a low Mach number ( $M \ll 1$ ), the flow is nearly incompressible, whereas for $M \ge 1$ , compressibility effects and shock waves become significant.

Aerodynamic Similarity

Using an aerodynamic example of flow similarity, consider two bodies to be tested in two wind tunnels to determine their relative aerodynamic performance. The bodies are said to be geometrically similar if the geometry of one body can be obtained by applying a single scale factor to the geometry of the other body, as shown in Figure 1. However, suppose the two bodies (e.g., a model and a full-scale application) are not geometrically similar. In that case, kinematic and dynamic flow similarity cannot be achieved, which is the end of the matter. Therefore, geometrical similarity between the two bodies is a prerequisite for achieving kinematic and dynamic flow similarity; hence, the use of geometrically scaled models in the wind tunnel is justified.

A prerequisite for flow similarity between two bodies is that a single scaling factor scales their geometries.

Suppose all similarity parameters applicable to each geometrically scaled body can be set equal, e.g., the same values of the Reynolds and Mach numbers. In that case, the flows about each body will be kinematically and dynamically similar, and the results obtained at the model scale will apply to the full-scale application. Therefore, in summary, it can be stated that the flows about two bodies will be dynamically similar if:

The body shapes are geometrically similar, i.e., a single scaling factor relates the model and full-scale shapes.
All similarity parameters, such as Reynolds number $Re$ , Mach number $M$ , and other relevant similarity parameters, have the same values.

In the case of aerodynamic problems, if it can be formally established that the flow similarity parameters have the same values for both flows at both scales and that the flows are dynamically similar, then:

The streamline patterns of both bodies will be geometrically similar. This means the flow patterns around both bodies will be the same.
The flow distributions of non-dimensional velocity, pressure coefficient, etc., will be the same when plotted against a standard non-dimensional length coordinate, i.e., a length scale non-dimensionalized by a characteristic length such as a chord.
The force coefficients (e.g., lift and drag coefficients) and moment coefficients (about the same non-dimensional reference point) will be the same.

The Mach number and Reynolds number are the most significant similarity parameters used in aerodynamics. Remember that the Reynolds number is always based on some characteristic length (e.g., chord, mean chord, diameter, etc.) and should be qualified as such. Other similarity parameters may be necessary for specific problems, especially when factors beyond aerodynamics are involved. However, the effects of Mach number and Reynolds number inevitably come up in all aerospace flight vehicle problems in one form or another.

Wind Tunnel Testing

The concept of dynamic flow similarity is fundamental to wind-tunnel testing of subscale models; see Figure 2. Suppose a subscale model of an actual full-size aircraft will be tested in a wind tunnel. In that case, the basic idea is to simulate free-flight conditions, i.e., the actual flight conditions in the wind tunnel. In this case, the flow produced will yield identical non-dimensional pressure distributions, lift, moments, and drag coefficients to those experienced by the actual aircraft during free flight. However, this outcome can only be true if the Reynolds and Mach numbers attained in the wind tunnel test are the same as those for free flight. Scaled models in a wind tunnel can help verify the aircraft design before tooling and construction begin. In most cases, this is a critical step in the development process of a new aircraft.

Testing a sub-scale model in a wind tunnel. The model is an MD-11.

Unfortunately, it is challenging to properly scale both the Reynolds and Mach numbers in wind tunnel tests, primarily because the test article is typically a smaller (i.e., sub-scale) version of the actual aircraft. While the issues can be avoided by using a wind tunnel large enough to test the full-scale flight vehicle, such a tunnel is rarely practical. The largest wind tunnel is the National Full-Scale Aerodynamics Complex at NASA Ames Research Center, which features a test section measuring 120 feet (36.6 meters) wide by 80 feet (24.4 meters) high. It can even accommodate actual aircraft with their engines running. However, it can only be tested at flow speeds up to approximately 120 knots. Therefore, most wind tunnel tests are performed with smaller or sub-scale models.

Challenges of Wind Tunnel Testing

During the design phase, it is necessary to conduct wind tunnel tests on any new aircraft to verify the predicted aerodynamic performance. For example, consider the design of a light aircraft. Assume that the full-scale aircraft is expected to cruise at a speed of about 135 mph ( ${V_{\infty}}$ is approximately 60 m s $^{-1}$ ). The aircraft’s wingspan is 14 m, and the mean wing chord is 1.5 m.

For these preceding conditions, the Reynolds number based on the mean wing chord will be

(10) $\begin{equation*} Re = \frac{\varrho \, V_{\infty} \, c}{\mu} = \frac{1.23 \times 60 \times 1.5}{18.15 \times 10^{-6}} = 6.10 \times 10^6 \end{equation*}$

Similarly, the flight (freestream) Mach number will be approximately

(11) $\begin{equation*} M = \frac{V_{\infty}}{a} = \frac{60}{330} = 0.18 \end{equation*}$

Therefore, this wing will operate in an essentially incompressible flow regime. The freestream Mach number is so low (less than 0.3) that compressibility effects on the aerodynamics are likely negligible. If both the model and full-scale flows remain well within this incompressible regime, exact Mach-number matching is generally unnecessary, and Reynolds-number matching becomes the principal aerodynamic requirement.

Suppose, for example, a scale model of this aircraft were to be tested in a wind tunnel with a 2.5-by-3.5 m working section (the region in which the model is tested). Therefore, the wingspan of the model must be less than 3 m. Ideally, the span and frontal dimensions should be limited to control blockage and wind-tunnel wall interference, while the model length along the tunnel axis is governed by the usable test-section length, support arrangement, and flow uniformity. In this case, a wingspan of about 2 m would be appropriate.

A prerequisite for dynamic flow similarity is that the model must be geometrically similar to the actual aircraft. This requirement implies that a 1/7-scale model would be necessary and could be manufactured using various methods. Now, if the tunnel wind speed is set to 60 m s $^{-1}$ , which is the full-scale flight speed ${V_{\infty}}$ , the Reynolds number based on the mean chord will be approximately

(12) $\begin{equation*} Re = \frac{\varrho \, V_{\infty} \, c}{\mu} = \frac{1.23 \times 60 \times 1.5/7}{18.15 \times 10^{-6}} = 0.87 \times 10^6 \end{equation*}$

It is immediately apparent that dynamic similarity cannot be achieved in this case because the Reynolds numbers between the wind tunnel test and the full-scale aircraft cannot be matched, even if the Mach number can be matched. This situation is called partial similarity.

What can be done here? Well, the wind speed could be increased. However, to get the required matching of the Reynolds numbers, the tunnel wind speed must be seven times 60 m s $^{-1}$ , which is 420 m s $^{-1}$ . A wind speed of 420 m s $^{-1}$ will yield a Mach number far too high, about 1.3. So, even though the Reynolds numbers could be hypothetically matched, the Mach numbers would be incorrect. Therefore, one significant practical difficulty in wind-tunnel testing of scale models is now apparent, at least in conventional wind tunnels.

One solution to this dilemma is to change the working fluid’s density, viscosity (or both). In this regard, hydrodynamic tests (performed in water) are sometimes conducted on models because the kinematic viscosity of water is significantly lower (approximately 1/15) than that of air. This means that smaller models tested at lower speeds in water can achieve a Reynolds number nearly as high as that of larger models tested in faster-moving air. However, scaling the Mach number is also essential, as the speed of sound in water is approximately 4 times that in air. Therefore, water, as a test medium, is not typically suitable for simulating the parameters of flight vehicles.

Compressed air, helium, or a refrigerant gas may allow higher-Reynolds-number testing of smaller models. For example, a pressurized wind tunnel could compress the air, increasing its density. However, this is a complex and costly option because it requires a specialized wind tunnel. Air density can also be increased by cooling, thereby decreasing viscosity. Cooling could also be a viable option for increasing the Reynolds number; however, the air must be cooled to very low temperatures to substantially reduce its viscosity. Therefore, it is necessary to use a cryogenic wind tunnel, which is very expensive to build and run.

The primary reason for using cryogenic temperatures is that they enable higher Reynolds numbers. The Reynolds number can be written as

(13) $\begin{equation*} Re = \frac{\varrho \, V \, c}{\mu} = \left( \frac{\varrho \, a}{\mu} \right) M \, c \end{equation*}$

Therefore, for a given Mach number and model size, then

(14) $\begin{equation*} Re \, \propto \, \left( \frac{\varrho_{\infty} \, a_{\infty}}{\mu_{\infty}} \right) \end{equation*}$

For an ideal gas, then

(15) $\begin{equation*} \varrho_{\infty} \, \propto \, \frac{p_{\infty}}{T_{\infty}} \quad \mbox{and} \quad a_{\infty} \, \propto \, \sqrt{T_{\infty}} \quad \mbox{and} \quad \mu_{\infty} \, \propto \, \sqrt{T_{\infty}} \end{equation*}$

so that

(16) $\begin{equation*} Re \, \propto \, \left( \frac{p_{\infty}}{T_{\infty}} \right) \left( \frac{T_{\infty}^{1/2}}{T_{\infty}^{1/2}} \right) \approx \frac{p_{\infty}}{T_{\infty}} \end{equation*}$

Therefore, the Reynolds number increases as the temperature decreases for a given Mach number, model size, and tunnel operating pressure.

Helpful Scaling Laws

Many fluid parameters appear as ratios in similarity analysis, including the speed of sound, viscosity, and density. These ratios can often be reexpressed in terms of temperature ratios, which is particularly helpful for establishing scaling factors and the relative values of similarity parameters, even if only approximately.

For example, the speed of sound is given by

(17) $\begin{equation*} a = \sqrt{ \gamma \, R \, T} \end{equation*}$

where $T$ is the absolute temperature. Therefore, in terms of the speed of sound ratio between two conditions, 1 and 2, then

(18) $\begin{equation*} \dfrac{a_1}{a_2} = \sqrt{ \frac{T_1}{T_2} } \end{equation*}$

The viscosity of a gas is approximately proportional to the square root of temperature, i.e., $\mu \, \propto \, \sqrt{T}$ , which is an outcome of the kinetic theory. Therefore, between the two conditions, 1 and 2, then

(19) $\begin{equation*} \dfrac{\mu_1}{\mu_2} \approx \sqrt{ \frac{T_1}{T_2} } \end{equation*}$

From thermodynamics, then $\varrho \, \propto \, 1/T$ at a constant pressure, so

(20) $\begin{equation*} \dfrac{\varrho_1}{\varrho_2} = \frac{T_2}{T_1} \end{equation*}$

Finally, the kinematic viscosity, $\nu (= \mu / \varrho) \, \propto \, T^{3/2}$ so that

(21) $\begin{equation*} \dfrac{\nu_1}{\nu_2}\approx \left( \frac{T_1}{T_2} \right)^{3/2} \end{equation*}$

While some of these preceding relationships are only approximate, they can be used to yield reasonable estimates of scaling effects across conditions and to elucidate how dynamic similarity may be achieved.

Check Your Understanding #1 – Confirming dynamic similarity

Consider two airfoils with the same profile shape and operating angle of attack but different chord lengths, operating in two different fluids, as shown in the table below. Determine whether or not the flows are dynamically similar.

$\small \mbox{\bf Airfoil 1}$

$\small \mbox{\bf Airfoil 2}$

$\small \alpha_1 = 5^{\circ}$

$\small\alpha_2 = 5^{\circ}$

$\small V_1 = 210~\mbox{m} \, \mbox{s}^{-1}$

$\small V_2 = 140~\mbox{m}\, \mbox{s}^{-1}$

$\small \varrho_1 = 1.2~\mbox{kg} \,\mbox{m}^{-3}$

$\small \varrho_2 = 3.0~\mbox{kg}\, \mbox{m}^{-3}$

$\small\mu_1 = 1.8 \times 10^{-5}~\mbox{kg m}^{-1}\mbox{ s}^{-1}$

$\small \mu_2 = 1.5 \times 10^{-5}~\mbox{kg m}^{-1}\mbox{s}^{-1}$

$\small a_1 = 300~\mbox{m} \, \mbox{s}^{-1}$

$\small a_2 = 200~\mbox{m} \, \mbox{s}^{-1}$

$\small c_1 = 1.0~\mbox{m}$

$\small c_2 = 0.5~\mbox{m}$

Show solution/hide solution.

The test for dynamic flow similarity requires determining the values of the similarity parameters, specifically the Reynolds number and Mach number, for each flow. For Airfoil 1, then for the Reynolds number

$Re_1 = \frac{\varrho_1 V_1 c_1}{\mu_1} = \frac{1.2 \times 210.0 \times 1.0}{1.8 \times 10^{-5}} = 1.4 \times 10^7$

and for the corresponding Mach number

$M_1 = \frac{V_1}{a_1} = \frac{210.0}{300.0} = 0.7$

For Airfoil 2, then for the Reynolds number

$Re_2 = \frac{\varrho_2 V_2 c_2}{\mu_2} = \frac{3.0 \times 140.0 \times 0.5}{1.5 \times 10^{-5}} = 1.4 \times 10^7$

and for the corresponding Mach number

$M_2 = \frac{V_2}{a_2} = \frac{140}{200.0} = 0.7$

Therefore, despite differences in airfoil size and flow conditions, these two flows are indeed dynamically similar.

Partial Similarity

In many cases of partial similarity, especially at lower Mach numbers, the aerodynamics are more significantly affected by the Reynolds number. Therefore, Reynolds-number matching alone may provide an acceptable approximation when compressibility effects are negligible. Sometimes, surface roughness is applied to the model to induce boundary-layer transition and produce flow separation at the location observed in the full-scale application. At higher Mach numbers, where compressibility effects are significant, Mach-number matching may be given priority while Reynolds-number mismatch is accepted as a source of partial similarity. In either case, complete dynamic similarity has not been achieved unless all relevant similarity parameters are matched.

Certain types of sensitivity analysis usually establish whether the effects of one or more similarity parameters can be marginalized or ignored in a given problem. However, it should never be assumed a priori that any similarity parameter that could affect the problem can be ignored. This cautionary note serves as a reminder of the importance of thorough analysis and the potential pitfalls of hasty assumptions.

Extrapolation methods can also be used when only partial similarity is attainable. The most common scenario in the wind tunnel is to match the Mach number, but the Reynolds number is at least an order of magnitude lower than the full-scale flight value. In Figure 3, for example, Reynolds-number scaling for measurements above approximately 2 million is satisfactory, indicating that the results from wind tunnel B are applicable. In this case, if the force coefficients at full-scale (fs) and model scale (ms) are $C_{F_{\rm fs}}$ and $C_{F_{\rm ms}}$ , respectively, then

(22) $\begin{equation*} C_{F_{\rm fs}} = \left( \frac{Re_{\rm fs}}{Re_{\rm ms}} \right)^{\alpha} C_{F_{\rm ms}} \end{equation*}$

where the value of ${\alpha}$ can be obtained from a power-law fit, often by applying linear least squares after taking logarithms. Extrapolation from measurements made in wind tunnel A would be considered unsatisfactory.

Reynolds-number scaling of aerodynamic coefficients can be a satisfactory approach, mainly when supported by CFD results.

However, discrepancies between wind-tunnel and full-scale flight measurements may have causes that extend beyond mismatches in similarity parameters. These causes may include model-shape fidelity, model-support interference, wall corrections, and aeroelastic effects, among others. Additionally, commonly used methods for simulating higher-Reynolds-number flows, such as artificially tripping the boundary layer, may lead to under- or overprediction of full-scale values. Current approaches to wind-tunnel data extrapolation use computational fluid dynamics (CFD) to validate the extrapolation process.

Similitude in Various Engineering Fields

As previously discussed, two systems are geometrically, kinematically, and dynamically similar when all similarity parameters have the same numerical values; this is used to verify that the problem’s scaling exhibits similitude. In practice, however, achieving complete similarity is difficult when disparate geometric scales are involved, i.e., a very small model compared to the full-scale article. Nevertheless, the principles of dynamic scaling are essential because many engineering design processes become established when testing smaller models of the full-scale system(s).

Similitude analysis is a powerful design tool across fields beyond aerodynamics. For example, scaling laws and similarity parameters can be derived across fields such as structures, structural dynamics, aeroelasticity, and hydrodynamics. The idea, again, is to scale the larger “real” or “full-scale” physical problem downward to a model and/or a smaller prototype. If applied correctly to match the similarity parameters governing the problem’s physics, it will yield the necessary similarity in physical behavior between the two disparate geometric scales.

However, just as in aerodynamics, designing a scaled-down structure (e.g., a wing or a complete flight vehicle) that matches all of the problem’s similarity parameters is very challenging. Nevertheless, relaxing one or more scaling parameters may be appropriate, thereby allowing at least partial similarity to a scaled model. To this end, parametric variations across specific scales can reveal the expected sensitivities of one parameter relative to another and highlight the more critical scaling parameter(s).

Structural & Aeroelastic Models

The behavior of an aircraft structure, such as a wing, depends on its stiffness, damping, and mass properties. In addition to force similitude, displacement similitude is usually enforced, so the model’s shape in the wind tunnel replicates that obtained during flight. For most flight vehicles that undergo relatively large deformations, inertial, gravitational, and restoring forces are critical to the design of a sub-scale model. Therefore, aeroelastically scaled models that incorporate coupled aerodynamic-structural scaling can be designed to replicate their full-scale counterparts.

In this case, scaling a model that is aeroelastically similar to an aircraft requires that its characteristics under steady loads match those of the full-scale aircraft. i.e., a statically aeroelastically scaled model deflects to the same shape and with a scaled magnitude under the same scaled static loads. The question of what material a model should be made of is also a consideration. For example, it may not be possible to replicate a wing model using the same construction methods and materials as those used for the full-scale wing.

The basic nondimensional relationships governing structural deformations can be derived from dimensional analysis. If a lift force $L$ is applied to a cantilevered wing of semi-span ${s}$ , second moment of area $I$ , and modulus of elasticity, $E$ , then the tip deflection, $z_t$ , can be written in general functional form as

(23) $\begin{equation*} \frac{z_t}{s} = \phi \left( \frac{ I}{s^4}, \frac{ L}{E \, s^2} \right) \end{equation*}$

The lift force on the wing is proportional to the freestream dynamic pressure, ${q_{\infty}}$ , and the wing reference (planform) area, $S$ , so the functional relationship can be reexpressed as

(24) $\begin{equation*} \frac{z_t}{s} = \phi \left( \frac{ I}{s^4}, \frac{ q_{\infty} \, S}{E \, s^2} \right) \end{equation*}$

Notice that because $s$ is the wing semi-span, $AR = 4s^2/S$ . The constant factor of 4 can be absorbed into the unspecified functional relationship, so

(25) $\begin{equation*} \frac{z_t}{s} = \phi \left( \frac{ I}{s^4}, \frac{ q_{\infty}}{E \, AR } \right) \end{equation*}$

Therefore, the relevant similarity parameters governing the aeroelastic deformations of the wing are $z_t/s$ , $I/s^4$ , and ${q_{\infty}S/(E \, s^2)}$ . For aeroelastic scaling, these results are usually combined such that dynamic similarity requires

(26) $\begin{equation*} \left( \frac{q_{\infty} \, s^4}{E \, I} \right)_{\rm ms} = \left( \frac{q_{\infty} \, s^4}{E \, I} \right)_{\rm fs} \end{equation*}$

where $E \, I$ is the wing stiffness or flexural rigidity. Therefore,

(27) $\begin{equation*} \left( \frac{(E \, I)_{\rm ms}}{(E \, I)_{\rm fs}}\right) = \left( \frac{q_{{\infty}_{\rm ms}}}{q_{{\infty}_{\rm fs}}}\right) \left( \frac{s_{\rm ms}}{s_{\rm fs}}\right)^4 \end{equation*}$

Maintaining similarity using Eq. 26 is required for faithful aeroelastic testing of a subscale model. For example, with a half-scale model being tested at the same dynamic pressure, the wing must have one-sixteenth of the flexural rigidity of the full-scale wing. For testing at lower dynamic pressures, which is often the case in wind tunnels, the wing stiffness must be reduced even further. As shown in Figure 4, the goal is to maintain kinematically similar wing deformations under the reduced aerodynamic loads. However, because such models are typically smaller than half-scale, the resulting model is often highly flexible and requires careful manufacturing and testing.

Aeroelastic testing of sub-scale wings requires reduced bending and torsional stiffness to match full-scale deformation behavior.

In addition, for an aeroelastic model such as a wing, the Froude number, $Fr$ , is usually relevant, which is defined as

(28) $\begin{equation*} Fr = \frac{V_{\infty}}{\sqrt{ s \, g}} \end{equation*}$

The Froude number relates the aerodynamic inertial forces to the effects of gravity, which should also be matched to ensure dynamic similarity when wing flexibility, weight, and aeroelastic effects are relevant, i.e.,

(29) $\begin{equation*} \left( \frac{V_{\infty}^2}{ s \, g} \right)_{\rm ms} = \left( \frac{V_{\infty}^2}{ s \, g} \right)_{\rm fs} \end{equation*}$

In this regard, the Froude number quantifies the relative importance of inertial effects and gravity; the resulting structural deflections also depend on stiffness, mass distribution, geometry, and loading.

Flutter Models

Flutter clearance is considered critical when developing new aircraft systems. As previously discussed, the interaction between aerodynamics and the structural response (often called structural dynamics) appears in terms of the ratio between forces induced by dynamic pressure relative to the structure’s stiffness for a given aspect ratio (required for geometric scaling), i.e., $\dfrac{q_{\infty}}{E}$ as well as mass density, which is the ratio of the structural mass, $M$ , to an appropriate volume grouping, i.e., $\frac{M}{s^3}$ . However, matching these ratios and similarity parameters typically yields a lighter, more flexible model for aeroelastic and flutter testing. It is not unusual for flutter models to have segmented wing sections so that the wing has the necessary low stiffness in bending and torsion. The model may also have to operate at cryogenic temperatures, which can be challenging to achieve in practice. Nevertheless, a significant advantage of the wind tunnel approach is that measurements can be made at a smaller and more convenient physical scale to understand the effects of the primary problem parameters, validate mathematical models, and identify any potentially undesirable, if not catastrophic, aeroelastic or flutter behavior that could manifest during the flight of the full-scale vehicle.

In Figure 5, a proprotor for a tiltrotor aircraft is mounted on a flexible, aeroelastically scaled wing to study the potential for whirl-flutter instability. This flutter is caused by coupling the proprotor aerodynamics and the proprotor’s dynamic response on the wing. The onset of whirl-flutter is often a limiting factor in a tiltrotor’s forward airspeed capability.

An example of a wing flutter model. In this case, a proprotor is mounted on a flexible, aeroelastically scaled wing.

Aerodynamic time scales are also essential in aeroelasticity, i.e., the need to simulate the non-dimensional time scales of the model and the full-scale application. In this regard, the reduced frequency may be relevant regarding oscillatory behavior. For structural dynamic similitude, the dimensionless parameters pertinent to the scaled model include the reduced frequency, typically denoted by $k$ , and the Froude number, $Fr$ . For an airfoil or wing section undergoing harmonic motion, the reduced frequency is usually defined as

(30) $\begin{equation*} k = \frac{\omega\, c}{2 V_{\infty}} \end{equation*}$

where $\omega$ is the circular frequency of the motion, $c$ is the chord, and $V_{\infty}$ is the freestream velocity. Equivalently, if the semi-chord is denoted by $b=c/2$ , then

(31) $\begin{equation*} k = \frac{\omega\, b}{V_{\infty}} \end{equation*}$

Care is therefore needed because the symbol $b$ is also often used for wing span or semi-span in other contexts.

However, for transient time response the reduced time, $\hat{s}$ , may be more relevant, i.e.,

(32) $\begin{equation*} \hat{s} = \frac{2 V_{\infty} \, t}{c} \end{equation*}$

where $t$ is time and ${c}$ is the wing chord. A physical interpretation of this non-dimensional time is the distance the wing travels through the flow in terms of semi-chord lengths.

Hydrodynamic Models

Designing a new ship, yacht, seaplane, or other vessel usually requires testing a sub-scale model in a towing tank. Tests may also be performed to improve the design of an existing or modified vehicle, for example, to enhance its performance by reducing the hydrodynamic drag on a ship as it moves through the water, as illustrated in Figure 6. The hydrodynamic drag on a ship’s hull is caused by both viscous effects (from viscous shear on the hull) and gravitational effects (from wave motion). In the latter case, a ship traveling over a sea leaves behind a train of waves, and because these waves possess energy that is eventually dissipated, the ship experiences a wave-drag force.

The hydrodynamic drag on a ship depends on its speed and wave height.

The drag coefficient on the hull of the ship can be written in functional form as

(33) $\begin{equation*} C_D = \phi (Re, Fr) \end{equation*}$

where $Re$ is the Reynolds number based on ship length, ${l}$ , and the dimensionless grouping

(34) $\begin{equation*} Fr= \frac{V}{\sqrt{g \, l}} \end{equation*}$

is the corresponding Froude number, where $V$ is the ship’s speed through the water. The resistance from viscous effects is a function of the Reynolds number and the roughness of the hull. In model ship testing, separating these two components (skin friction or shear drag and wave-wake drag) enables determination of the hydrodynamic drag of actual ships from tests conducted with smaller models, typically at 1:25 or 1:50 scale, such as in towing tanks; see Figure 7.

Ship drag can be estimated using a scaled model tested at a matching Froude number, with viscous-drag corrections applied for the Reynolds-number difference.

If the hydrodynamic Froude number is much less than unity, implying that the waves’ wavelength is smaller than the ship’s length, then the resulting wave drag on the hull is comparatively small. If the wavelength of the generated waves becomes comparable to the ship’s length, the bow and stern wave systems interfere strongly, resulting in the so-called “hump speed,” which leads to a more significant value of wave drag. Therefore, a heavy ship with large volumetric displacement generally cannot overcome this peak in the wave drag, so its cruise speed will be limited.

High-speed boats, which can reach Froude numbers exceeding 3, can experience different regimes and eventually enter the so-called planing regime, in which they skim over the water with significantly reduced drag. The same behavior occurs during the takeoff run of a seaplane, when its weight is eventually supported by hydrodynamic lift rather than buoyancy, a condition known as operating “on the step.” Indeed, the seaplane’s ability to take off from the water and fly depends on the pilot’s ability to reach this low-drag hydrodynamic condition.

Check Your Understanding #2 – Reynolds & Mach numbers for a rocket

Estimate the Mach and Reynolds numbers (based on length) for a proposed launch vehicle near “max-q.” This condition corresponds to a flight speed of 450 m/s at an altitude of 34,000 ft. The actual launch vehicle’s characteristic length is 134 ft. Next, considering the $Re$ and $M$ Maps for the NASA Unitary Plan Wind Tunnel, as shown below (click the image for a larger version), can these flight conditions be simulated in this wind tunnel using a subscale model of the launch vehicle? What scale might you recommend for the wind tunnel model, and why? Which test section would you recommend? What might be done if the flight conditions cannot be replicated in this wind tunnel?

Show solution/hide solution.

At an altitude of 34,000 ft, the ISA standard values of the air properties are density $\varrho$ = 0.00076706 slugs ft ${^{-3}}$ , dynamic viscosity $\mu$ = 3.017 $\times 10^{-7}$ slugs ft $^{-1}$ s $^{-1}$ , speed of sound $a$ = 977.52 ft s $^{-1}$ . A flight speed $V$ of 450 m/s is 1,476.38 ft/s, so the flight Mach number is

$M = \frac{1,476.38}{977.53} = 1.51$

The corresponding Reynolds number based on the characteristic length is

$Re = \frac{\varrho \ V \ l}{\mu} = \frac{0.00076706 \times 1,476.38 \times 134.0}{3.017 \times 10^{-7}} = 5.03 \times 10^8$

or in terms of Reynolds number per foot, then

$Re = \frac{\varrho \ V }{\mu} = \frac{0.00076706 \times 1,476.38 }{3.017 \times 10^{-7}} = 3.75 \times 10^6~\mbox{per foot}$

Examining the $Re$ and $M$ Maps for the NASA Unitary Plan Wind Tunnel, the 9-foot-by-7-foot supersonic test section will give the required Mach number. The other transonic test section can reach a Mach number of up to 1.4. The 9-foot-by-7-foot test section can also reach a Reynolds number of about 5 million per foot.

For a wind tunnel model, the allowable scale is governed by the usable test-section length, support arrangement, flow quality, and the model’s cross-sectional blockage. Because this launch vehicle is slender and aligned with the flow, its length need not be limited by the smaller cross-sectional dimension of the tunnel. A practical model scale should be selected from the facility’s usable test-section dimensions and allowable blockage. For example, a model about 7 ft long would have a Reynolds number of approximately 35 million at a Mach number of 1.51, which is still about one order of magnitude lower than the actual flight Reynolds number.

When simulating the aerodynamic conditions of supersonic flight, matching the Mach number is usually prioritized. In this case, it is possible to achieve the correct flight Mach number in the wind tunnel. However, because the Reynolds number is still lower, the results should be regarded as an approximation unless Reynolds-number sensitivity is shown to be sufficiently small or is corrected using additional tests or CFD. This is another example of the challenges of sub-scale testing for studying fundamental problems. However, with some ingenuity, the problem can be studied by matching the similarity parameters that govern the physics as closely as possible.

Rotating & Propulsive System Similarity

The similarity of rotating systems, such as propellers, rotors, and turbomachinery, is inherently more complex than that of fixed lifting surfaces because the flow is characterized by multiple velocity scales and, in many cases, intrinsic unsteadiness. The local flow experienced by a blade is not determined solely by the freestream velocity, but by the superposition of the freestream and the rotational velocity, which varies along the span. Consequently, no single velocity scale is sufficient to describe the flow.

In forward flight, the ratio between the freestream velocity and the rotational speed is expressed through the advance ratio, i.e.,

(35) $\begin{equation*} J = \frac{V_{\infty}}{n \, D} \end{equation*}$

where ${V_{\infty}}$ is the freestream velocity, $n$ is the rotational speed in revolutions per second, and $D$ is the propeller or rotor diameter. This parameter governs the inflow kinematics and determines the propulsor’s operating state.

SR-3 High-Speed Turboprop rig and Laser Doppler Velocimeter in NASA’s Supersonic Wind Tunnel.

At the same time, compressibility effects are controlled not solely by the freestream Mach number but also by the local blade speeds, which are highest near the tip. This effect is expressed through the tip Mach number,

(36) $\begin{equation*} M_{\rm tip} = \frac{\Omega R}{a} \end{equation*}$

where $\Omega$ is the angular velocity, $R$ is the blade radius, and $a$ is the speed of sound. Even when the freestream Mach number is low, the blade-tip speed $\Omega R$ may approach transonic conditions.

The viscous scaling is equally problematic. A Reynolds number based on a blade section may be written as

(37) $\begin{equation*} Re = \frac{\varrho \, V_{\rm local} c}{\mu} \end{equation*}$

where $V_{\rm local}$ is the local flow speed seen by the blade section, $c$ is the local chord, $\varrho$ is the fluid density, and $\mu$ is the dynamic viscosity. Because $V_{\rm local}$ varies with radial position along the blade, the Reynolds number is not uniform, leading to spanwise variations in boundary-layer behavior, separation, and profile drag.

These complications are further compounded when unsteady or aeroelastic effects are present. The reduced frequency,

(38) $\begin{equation*} { k = \frac{\omega \, c}{2 \, V_{\infty}} } \end{equation*}$

where $\omega$ is a characteristic oscillation frequency, introduces a time scale into the problem. In addition, parameters such as the Lock number may be relevant, i.e.,

(39) $\begin{equation*} \gamma = \frac{\varrho \, a \, c \, R^4}{I_{\beta}} \end{equation*}$

where $a$ is the blade-section lift-curve slope and $I_{\beta}$ is the blade flapping moment of inertia. The Lock number characterizes the coupling between aerodynamic loading and blade inertia.

The essential difficulty is that these parameters impose conflicting requirements on the scaling of velocity, rotational speed, density, and geometry. Matching the tip Mach number fixes the angular velocity, whereas matching the Reynolds number may require a different velocity scale or fluid properties. At the same time, maintaining the correct advance ratio constrains the relationship between freestream speed and rotation rate. These requirements cannot, in general, be satisfied simultaneously.

For this reason, similarity in rotating systems is rarely exact. Instead, it is established in a controlled, deliberate manner by prioritizing the parameters that govern the dominant physics. In propeller testing, for example, the advance ratio and compressibility effects are often preserved, while Reynolds number effects are accepted as a source of distortion. The interpretation of such results, therefore, depends not only on the measurements themselves but on an understanding of which aspects of the physics have been faithfully reproduced and which have not.

Rarefied & High-Altitude Flow Similarity

The concept of dynamic similarity, as developed from the Navier-Stokes equations, implicitly assumes that the flow behaves as a continuum. This assumption is valid when the characteristic length scale of the problem is much larger than the molecules’ mean free path. However, at sufficiently high altitudes or for sufficiently small bodies, this assumption begins to fail, and the flow must instead be described in terms of molecular interactions.

As previously discussed, the relevant similarity parameter in this case is the Knudsen number, defined as

(40) $\begin{equation*} Kn = \frac{\lambda}{L} \end{equation*}$

where $\lambda$ is the molecular mean free path and $L$ is a characteristic length scale. When $Kn$ is very small, intermolecular collisions dominate, and the continuum approximation is appropriate. As $Kn$ increases, the influence of the gas’s molecular structure becomes increasingly important, leading to slip at solid boundaries, non-equilibrium effects within the flow, and ultimately to a regime in which the motion of individual molecules governs the aerodynamic behavior.

Contoured nozzle of the hypersonic wind tunnel H2K with the ExoMars-Schiaparelli model.

The implications for similarity are fundamental. Matching the Reynolds and Mach numbers between two flows ensures similarity only within the continuum framework. If the Knudsen numbers differ, then the governing physics is no longer the same, regardless of how well the classical similarity parameters are matched. In this sense, the Knudsen number is both a similarity parameter for rarefied flows and a measure of the validity of the continuum description itself. When the Knudsen number becomes sufficiently large, continuum-based methods must be replaced by molecular approaches, such as direct simulation Monte Carlo methods, which explicitly account for particle interactions. In this regime, similarity must be interpreted in terms of molecular statistics rather than in terms of continuum field variables.

This issue arises in several important contexts, including high-altitude flight, hypersonic vehicles, and small-scale aerodynamic systems. It is also encountered in wind-tunnel testing when the model scale becomes sufficiently small that the ratio of mean free path to characteristic dimension is no longer negligible. In such cases, experimental results may depart from full-scale behavior in ways that cannot be corrected through conventional scaling arguments.

The influence of density in these regimes can also be viewed through the mass-to-displaced-fluid parameter, as given by

(41) $\begin{equation*} Le = \frac{m}{\varrho \, L^3} \end{equation*}$

which increases as the ambient density decreases. At high altitudes, where density is low, the body’s inertia becomes increasingly dominant relative to the surrounding fluid. In this limit, the aerodynamic forces become comparatively weaker, and the motion of the vehicle approaches that of a ballistic body rather than one strongly coupled to the flow. This effect is distinct from rarefaction, which is governed by the Knudsen number. While the Knudsen number determines the validity of the continuum description, the Leishman parameter $Le$ governs the relative importance of fluid forces in determining the motion of the body. Both effects may become important simultaneously in low-density environments and must be considered independently when interpreting flight behavior.

Thermal & Energy Similarity

The similarity relationships developed from the momentum equation describe the scaling of forces and motion, but they do not, by themselves, ensure similarity of thermal fields. In many aerodynamic problems, particularly at high speeds or in propulsion systems, the transport of energy and the resulting temperature distributions play a central role in determining the overall flow behavior.

Thermal similarity introduces additional parameters that describe the relative rates of momentum and energy transport. The Prandtl number,

(42) $\begin{equation*} Pr = \frac{\mu \, c_p}{K} \end{equation*}$

relates the diffusion of momentum to that of heat and therefore governs the structure of the thermal boundary layer relative to the velocity boundary layer. The Nusselt number commonly characterizes convective heat transfer,

(43) $\begin{equation*} Nu = \frac{h \, L}{K} \end{equation*}$

while the Peclet number represents the combined effects of convection and diffusion, i.e.,

(44) $\begin{equation*} Pe = Re \, Pr \end{equation*}$

which provides a measure of the relative importance of advective and diffusive thermal transport.

The rate at which heat is transferred between the fluid and a surface may also be expressed through the Stanton number, denoted here by $St_h$ , i.e.,

(45) $\begin{equation*} St_h = \frac{h}{\varrho_{\infty} \, V_{\infty} \, c_p} \end{equation*}$

and when heat conduction within a solid body is important, the Biot number, i.e.,

(46) $\begin{equation*} Bi = \frac{h \, L}{K_s} \end{equation*}$

becomes relevant in determining the internal temperature gradients.

The key point is that these parameters introduce additional thermal-similarity requirements beyond matching the Reynolds and Mach numbers. Consequently, two flows that are dynamically similar in terms of force scaling may exhibit very different temperature fields, heat transfer rates, and thermal loads. This distinction is particularly important in high-speed flows, where aerodynamic heating, shock-layer temperatures, and viscous dissipation can strongly influence both the fluid and the structure.

In practical terms, this means that conventional wind tunnels, which are designed primarily to reproduce the correct velocity and pressure fields, may not reproduce the correct thermal environment. Specialized facilities, such as high-enthalpy tunnels or arc-jet test systems, are often required to simulate the combined effects of high speed and high temperature. Even in these facilities, achieving complete thermal similarity remains a significant challenge.

Therefore, whenever temperature, heat transfer, or energy exchange is integral to the problem, similarity must be assessed not only in terms of momentum scaling but also in terms of the additional parameters governing thermal transport. Failure to do so can lead to substantial errors in predicting material response, boundary-layer behavior, and overall system performance.

Summary of Similarity Parameters

Remember that similarity parameters quantify and analyze the physical similarity between systems, models, and full-scale articles. These parameters enable engineers to understand how the behavior of one system, such as a model, relates to that of another, often allowing the prediction or analysis of physical phenomena across different scales. Below is a summary of commonly used similarity parameters in aerospace engineering, along with their definitions.

In the definitions below, $V$ denotes a reference velocity appropriate to the physics being modeled. For external aerodynamic flows, this velocity is typically the freestream velocity, ${V_{\infty}}$ . However, in rotating systems, local blade-section flows and heat-transfer problems may require other velocity scales, such as the local resultant velocity $V_{\rm local}$ or the blade speed $\Omega R$ . Therefore, $V$ should always be interpreted as the relevant reference velocity for the problem, not as a universal velocity scale.

1. Reynolds number, $Re$

$Re = \frac{\varrho \, V \, L}{\mu}$

where $\varrho$ is the fluid density, $V$ is the selected reference velocity (often ${V_{\infty}}$ in external flows), $L$ is a characteristic length, and $\mu$ is the dynamic viscosity. The Reynolds number measures the ratio of inertial to viscous forces and governs boundary-layer behavior, flow separation, and transition.

2. Mach number, $M$

$M = \frac{V}{a}$

where $V$ is the relevant flow speed (typically ${V_{\infty}}$ in external flows) and $a$ is the speed of sound. The Mach number characterizes compressibility effects and determines whether the flow is subsonic, transonic, supersonic, or hypersonic.

3. Froude number, $Fr$

$Fr = \frac{V}{\sqrt{g \, L}}$

where $g$ is gravitational acceleration. This parameter measures the ratio of inertial to gravitational forces and is important in free-surface flows, flight dynamics, and aeroelastic scaling.

4. Weber number, $We$

$We = \frac{\varrho \, V^2 \, L}{\sigma}$

where $\sigma$ is the surface tension. The Weber number quantifies the relative importance of inertial forces compared with capillary effects and is relevant in multiphase flows and atomization.

5. Strouhal number, $St$

$St = \frac{f \, L}{V}$

where $\scriptstyle f$ is a characteristic frequency and $V$ is the reference velocity. This parameter describes unsteady flow phenomena such as vortex shedding and oscillatory aerodynamic behavior.

6. Reduced frequency, $k$

$k = \frac{\omega \, c}{2 \, V}$

where $\omega$ is the circular frequency, $c$ is a reference length such as a wing chord, and $V$ is usually taken as the freestream velocity ${V_{\infty}}$ in aerodynamic applications. The reduced frequency governs unsteady aerodynamic response and aeroelastic effects.

7. Womersley number, $\alpha$

$\alpha = L \sqrt{\frac{\omega}{\nu}}$

where $\omega$ is the angular frequency and $\nu$ is the kinematic viscosity. The Womersley number characterizes oscillatory viscous flows and measures the relative importance of unsteady inertial effects compared with viscous diffusion.

8. Stokes number, $St_k$

$St_k = \frac{\tau_p \, V}{L}$

where $\tau_p$ is a characteristic particle relaxation time and $V$ is the reference flow velocity. The Stokes number quantifies a particle’s ability to follow the flow. For $St_k \ll 1$ , particles closely follow the fluid motion, whereas for $St_k \gg 1$ , particle inertia dominates and the particles decouple from the flow.

9. Euler number, $Eu$

$Eu = \frac{p}{\varrho \, V^2}$

where $p$ is a characteristic pressure. This parameter represents the ratio of pressure forces to inertial forces and is useful in compressible-flow analysis.

10. Advance ratio, $J$

$J = \frac{V_{\infty}}{n \, D}$

where $n$ is the rotational speed in revolutions per second and $D$ is the propeller or rotor diameter. The advance ratio governs the inflow state of rotating propulsive systems.

11. Tip Mach number, $M_{\rm tip}$

$M_{\rm tip} = \frac{\Omega \, R}{a}$

where $\Omega$ is the angular velocity and $R$ is the rotor radius. This parameter reflects compressibility effects based on blade-tip speed rather than freestream velocity.

12. Lock number, $\gamma$

$\gamma = \frac{\varrho \, a \, c \, R^4}{I_{\beta}}$

where $a$ is the blade-section lift-curve slope, $c$ is a representative blade chord, $R$ is the blade radius, and $I_{\beta}$ is the blade flapping moment of inertia. The Lock number measures the ratio of aerodynamic forcing to blade inertial resistance.

13. Prandtl number, $Pr$

$Pr = \frac{\mu \, c_p}{K}$

where ${c_p}$ is the specific heat at constant pressure and $K$ is the thermal conductivity. The Prandtl number measures the relative diffusion of momentum and heat.

14. Nusselt number, $Nu$

$Nu = \frac{h \, L}{K}$

where $h$ is the convective heat-transfer coefficient. The Nusselt number characterizes convective heat transfer relative to thermal conduction.

15. Péclet number, ${Pe}$

$Pe = Re \, Pr$

The Péclet number quantifies the relative importance of convective transport relative to thermal diffusion.

16. Stanton number, $St_h$

$St_h = \frac{h}{\varrho \, V \, c_p}$

where $h$ is the convective heat-transfer coefficient and $V$ is the reference velocity associated with convective transport. The subscript $h$ is used to distinguish this parameter from the Strouhal number.

17. Biot number, $Bi$

$Bi = \frac{h \, L}{K_s}$

where $K_s$ is the thermal conductivity of the solid. The Biot number indicates whether significant internal temperature gradients develop within a solid body.

18. Grashof number, $Gr$

$Gr = \frac{g \, \beta \, (T_s - T_\infty) \, L^3}{\nu^2}$

where $\beta$ is the thermal expansion coefficient and $\nu$ is the kinematic viscosity. The Grashof number characterizes buoyancy-driven flows and natural convection.

19. Knudsen number, $Kn$

$Kn = \frac{\lambda}{L}$

where $\lambda$ is the molecular mean free path. The Knudsen number determines the validity of the continuum assumption and becomes important in rarefied and high-altitude flows.

20. Leishman number, $Le$

$Le = \frac{m}{\varrho \, L^3}$

where $m$ is the mass of the body. This parameter represents the ratio of body inertia to the mass of the displaced fluid. For $Le \sim O(1)$ , the body is strongly coupled to the surrounding fluid, whereas for large values of $Le$ , such as in low-density environments, the motion approaches that of a ballistic body with relatively weaker aerodynamic influence.

These parameters are not independent in practice, and it is rarely possible to match all of them simultaneously in a scaled experiment. The appropriate selection of similarity parameters depends on the problem’s dominant physics and requires engineering judgment.

Summary & Closure

The principles encompassing geometric, kinematic, and dynamic similarity are used in all branches and disciplines of engineering. While developing appropriate similarity parameters for specific problems can be time-consuming, the benefits for the engineer are usually significant. Similarity parameters not only help generalize the system’s behavior but also enable the engineer to reduce the problem scope by minimizing the number of independent parameters considered in the design process, particularly when many are present. Therefore, similarity parameters can facilitate the transfer of data and results from one scale (e.g., a smaller scale) to another (e.g., a full-scale application).

Utilizing similarity parameters and the principles of dynamic similarity is a crucial tool in designing and optimizing engineering systems across various industries. One significant benefit of using and matching similarity parameters is the ability to replicate a particular physical behavior at a smaller scale, where it can be studied more carefully, e.g., by testing in a wind tunnel. For example, a physical model or wind tunnel test can be used to study the aerodynamics of a full-scale aircraft, and the results can then be scaled to the actual conditions the aircraft will encounter in flight. However, scaling flow properties, especially the Reynolds number, can be challenging at smaller scales. One solution is to use another fluid (or gas) in the wind tunnel. However, this approach presents several practical challenges, including the need to use a specialized wind tunnel to contain the gas. Nevertheless, this approach enables engineers to obtain valuable information and predict the aircraft’s behavior without building prototypes or conducting expensive full-scale flight tests.

While the use of similarity parameters is not limited to wind tunnels, testing scaled models of aerospace systems can be highly valuable in the engineering design process. This approach can help identify potential problems that can be addressed at an early stage, rather than being discovered when the actual system is deployed in the field. For example, predicting flutter in flight vehicles is essential to ensure the structural design is both stiff and lightweight.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Discuss some challenges of using a gas other than air to achieve dynamic flow similarity with a scale model in a wind tunnel.
Conduct research to determine the purpose of a cryogenic wind tunnel in testing scaled models.
Discuss how computational methods, such as computational fluid dynamics (CFD), might achieve similitude in virtual environments, reducing the need for physical models.
What is “flutter”? Do some research to determine why the potential for flutter can be a severe problem in airframe design.
Aeroelastic models are sometimes referred to as “Froude-scaled models.” What might be the governing similarity parameter in this case?
Emphasize the importance of dynamic similarity in engineering design and safety assessments, where understanding how a system behaves under different conditions is crucial.

Other Useful Online Resources about Dynamic Similarity

Wikipedia pages on similitude and dynamic similarity.
A video on dynamic similarity in fluid dynamics.
A video lecture reviewing dimensional analysis and similitude.
A video lecture from the University of Colorado on similitude and scaling.
This is an excellent video on what causes flutter.
Archival film from NASA Langley Research Center’s 16-Foot Transonic Dynamics Tunnel. Composite of several tests on propeller whirl and flutter of helicopter blades.
Video of small-scale tiltrotor whirl flutter tests.

License

Icon for the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License

Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Digital Object Identifier (DOI)

https://doi.org/https://doi.org/10.15394/eaglepub.2022.1066.n14