15 Dynamic Similarity

Introduction

The principle of dynamic similarity is connected to the preceding concepts used in dimensional analysis. By matching the similarity parameters between two different circumstances (e.g., in two or more different experiments), it can be ensured that the physics of both situations are correctly scaled and have the correct physical similarity. For example, suppose the relevant similarity parameters can be matched. In that case, engineers can usually study the physical characteristics of a large-scale system (and perhaps one still in the design process) at a smaller scale and under the controlled environment of the laboratory or the wind tunnel.

As previously discussed, the key similarity parameters used in aerodynamics are the Reynolds number and Mach number. However, in general, many other relevant similarity parameters arise in aerodynamics and engineering problem-solving, such as Froude number, reduced frequency, reduced time, Weber number, Strouhal number, Prandtl number, etc. Similarity parameters of various types are used in all fields of engineering. Therefore, understanding the principles associated with dynamic similarity and applying them is very important in engineering analyses.

Learning Objectives

  • Understand the concept of dynamic similarity and appreciate its significance as a tool for engineering problem-solving.
  • Use dimensional analysis to understand the scaling of problem parameters to achieve dynamic similarity, i.e., how the problem scales with geometrical size, operating conditions, etc.
  • Appreciate the challenges of using scaled models in wind tunnel testing to obtain aerodynamic measurements that would apply to full-scale flight articles.

Aerodynamic Similarity

Using an aerodynamic example of 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 can be obtained by applying a single scale factor to the geometry of the other. However, suppose the two bodies (e.g., model versus the full-scale application) are not geometrically similar. In that case, dynamic flow similarity can never be obtained, which is the end of the matter.

This means, therefore, that the geometrical similarity of the two bodies is a prerequisite to obtaining flow similarity, hence the idea of correctly geometrically scaled models, e.g., for use in the wind tunnel. If the flow similarity parameters applicable to each geometrically scaled body can be made equal, then the flows about each body will be dynamically similar.

The following formal criteria are required to achieve flow similarity or so-called similitude between the model scale and the eventual application:

  • Geometric similarity: The model is scaled correctly and has the same geometrical shape as the application, i.e., a single scaling parameter can be used to relate them.
  • Kinematic similarity: This means that displacements and velocities of both the model and application must be the same.
  • Dynamic similarity: The ratios of all forces in the model and the application are the same, i.e., the inertial forces, gravitational forces, viscous forces, pressure forces, elastic forces, surface forces, etc., must all be the same.
A prerequisite for flow similarity about two bodies is that the geometries on the bodies be scaled by a single scaling factor.

Therefore, in summary, it can be stated that the flows about two bodies will be kinematically and dynamically similar if:

  • The body shapes are geometrically similar, i.e., related by a single scaling factor.
  • All of the  similarity parameters such as Reynolds number Re, Mach number M, etc., are the same.

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

Remember that the similarity parameters Mach number and Reynolds number, which have already been introduced, are by far the most significant similarity parameters used in aerodynamics. Other similarity parameters may be important for certain problems, especially when other than 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 a fundamental issue in wind tunnel testing of sub-scale models. Suppose a sub-scale model of an aircraft is to be tested in a wind tunnel. In that case, the basic idea is to simulate the conditions of free flight, i.e., to simulate the actual flight conditions in the wind tunnel. In this case, the flow produced will give the same non-dimensional pressure distributions, as well as lift, moments, and drag coefficients, as on the actual aircraft during a free flight. However, this outcome can only be true if the Reynolds numbers and Mach numbers that are attained in the wind tunnel test are the same as for free flight. Scaled models in a wind tunnel can help verify the aircraft design before tooling and construction begin, and in most cases, it is a critical step in the development process of a new aircraft.

Testing a sub-scale model in a wind tunnel. In this case, the model is an MD-11.

Unfortunately, it is difficult to obtain the proper scaling of both the Reynolds number and the Mach number in wind tunnel tests mainly because the test article in the wind tunnel is usually a smaller (i.e., sub-scale) version of the actual aircraft. While the issues can be obviated (perhaps) by using a wind tunnel big enough to test the full-scale flight vehicle, although this is rarely practical. The biggest wind tunnel is the National Full-Scale Aerodynamics Testing Facility at NASA Ames Research Center, which has a test section 120 ft (36.6 m) wide and 80 ft (24.4 m) high. It can even accommodate actual aircraft with their engines running. However, it can only test at flow speeds up to about 120 knots. Therefore, most wind tunnel tests are performed with smaller or sub-scale models.

Challenges of Wind Tunnel Testing

During design, it is necessary to conduct wind tunnel tests on any new aircraft to verify 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 140 mph (V_{\infty} is approximately 50 ms^{-1}). In addition, the wing span of the aircraft 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

(1)   \begin{equation*} Re = \frac{\varrho V_{\infty} c}{\mu} = \frac{1.23 \times 50 \times 1.5}{18.15 \times 10^{-6}} = 5.08 \times 10^6 \end{equation*}

Similarly, the flight (free-stream) Mach number will be approximately

(2)   \begin{equation*} M = \frac{V_{\infty}}{a} = \frac{50}{330} = 0.16 \end{equation*}

Therefore, this wing will operate in an essentially incompressible flow regime, i.e., the free-stream Mach number is so low (less than 0.3) that compressibility effects on the aerodynamics are likely negligible. Nevertheless, this does not obviate the need to match the Mach number as one of the required similarity parameters.

Suppose, for example, a scale model of this aircraft was to be tested in a wind tunnel, the tunnel having a 2.5-by-3.5 m working section (the region where the model is actually tested). Therefore, the wingspan of the model must be less than 3 m. Ideally, the wing span should be between 50% and 70% of the maximum tunnel width, in this case, about 2 m. This size is essential to ensure that the wind tunnel walls do not disturb the flow over the wing; this is called a wind tunnel wall interference effect, and the effect should be minimized as much as possible.

A prerequisite for the dynamic flow similarity of the model is that the model must be geometrically similar to the actual aircraft. This requirement means something of the order of a 1/7-scale model would be needed. Now, if the tunnel wind speed is set to 50 ms^{-1} (which is the full-scale flight speed V_{\infty}), the Reynolds number based on the mean chord will be approximately

(3)   \begin{equation*} Re = \frac{\varrho V_{\infty} c}{\mu} = \frac{1.23 \times 50 \times 1.5/7}{18.15 \times 10^{-6}} = 0.72 \times 10^6 \end{equation*}

So, it can be seen immediately that dynamic similarity cannot be obtained in this case because the Reynolds numbers between the wind tunnel test and full scale cannot be matched, even if the Mach number can be matched.

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 50 ms^{-1}, which is 350 ms^{-1}, and a wind speed of 350 ms^{-1} will give us a Mach number that is far too high. So, even though the Reynolds numbers could be hypothetically matched, now the Mach numbers would be incorrect. Therefore, one of the significant practical difficulties of wind tunnel testing of scale models is now apparent, at least using conventional wind tunnels.

One solution to this dilemma is to change the density or viscosity (or both) of the working fluid. In this regard, hydrodynamic tests (in water) are sometimes performed on models because the kinematic viscosity of water is significantly less (about 1/15) than that of air. This means that smaller models tested at lower speeds in water can achieve almost the same Reynolds number as larger ones tested in faster-moving air. However, the issue of scaling the Mach number is also essential, bearing in mind that the ratio of the speed of sound in water compared to air is about 4:1.

Compressed air and liquid helium may allow higher-Reynolds number testing of smaller models. For example, a pressurized wind tunnel could be used to compress the air and increase its density. However, this is a complicated and expensive option because a special wind tunnel must be used. The density of the air could also be increased by cooling, which will, at the same time, decrease the viscosity. This air cooling could also be a viable option to increase the Reynolds number. Unfortunately, the air must be cooled to low temperatures to change viscosity. Therefore, it is necessary to use a special cryogenic wind tunnel, which is very expensive to build and run. Another solution to increase the Reynolds number is to use another fluid (or gas). However, this approach also has practical challenges, including using a special wind tunnel to contain the gas.

However, in many cases, at lower Mach numbers, the aerodynamics are affected more by the Reynolds number, so to obtain dynamic flow similarity, then Reynolds number matching alone may be sufficient. Sometimes surface roughness is applied to the model to force boundary layer transition and create the onset of flow separation at the exact location as the full-scale application. At higher Mach numbers, where compressibility effects are significant, the effects of the Reynolds number are less critical. In this case, only the Mach Number needs to be matched. Establishing whether the effects of one or more similarity parameters can be ignored in a given problem is usually accomplished by certain types of sensitivity analysis. It should never be automatically assumed that any similarity parameter potentially affecting the problem can be ignored a priori.

Example #1

Consider the flow about two geometrically similar airfoils, with one having four times the chord of the other airfoil, as shown in the figure below. The free-stream parameters are very different such that:

    \[ V_2 = 2 V_1, \quad \varrho_2 = \frac{ \varrho_1 }{4}, \quad \mbox{~and~} \quad T_2 = 4 T_1 \]

Are the flows dynamically similar? This outcome is certainly not obvious!

To find out we need to determine if the flow similarity parameters are the same for both flows, i.e., specifically to determine if

Re_{1} = Re_{2} and M_{1} = M_{2}. For the first flow we have

    \[ Re_{1} = \frac{\varrho_{1} V_{1} c}{\mu_{1}} \]

and for the second flow we have

    \[ Re_{2} = \frac{\varrho_{2} V_{2} 4c}{\mu_{2}} = \frac{\varrho_{1}}{4} \frac{2V_{1}4c}{\mu_{2}} = \frac{2\varrho_{1}V_{1}}{\mu_{2}} \]

Therefore,

    \[ \frac{Re_{1}}{Re_{2}} = \frac{\varrho_{1}V_{1}c}{\mu_{1}} \frac{\mu_{2}}{2\varrho_{1}V_{1}c} = \frac{\mu_{2}}{2\mu_{1}} \]

For a gas then \mu \propto \sqrt{T} is a good approximation so that

    \[ \frac{\mu_{1}}{\mu_{2}} = \frac{\sqrt{T_{1}}}{\sqrt{T_{2}}} \]

Therefore,

    \[ \frac{Re_{1}}{Re_{2}} = \frac{\mu_{2}}{2\mu_{1}} = \frac{\sqrt{T_{2}}}{2\sqrt{T_{1}}}= \frac{\sqrt{4T_{1}}}{2\sqrt{T_{1}}} = \frac{2\sqrt{T_{1}}}{2\sqrt{T_{1}}} = 1 \]

and so

    \[ Re_{1} = Re_{2} \]

So we have shown that the Reynolds numbers for both are equal.

The Mach numbers for the two flows are given by

    \[ M_{1} = \frac{V_{1}}{a_{1}} \mbox{ \quad and \quad } M_{2} = \frac{V_{2}}{a_{2}} = \frac{2V_{1}}{a_{2}} \]

This gives

    \[ \frac{M_{1}}{M_{2}} = \frac{a_{2}}{2a_{1}} \]

We also know that a\propto \sqrt{T}, so

    \[ \frac{a_{1}}{a_{2}} = \frac{\sqrt{T_{1}}}{\sqrt{T_{2}}} \]

and so

    \[ \frac{M_{1}}{M_{2}} = \frac{a_{2}}{2a_{1}} = \frac{\sqrt{T_2}}{2\sqrt{T_2}}=\frac{\sqrt{4T_1}} {2\sqrt{T_2}} = \frac{2\sqrt{T_1}}{2\sqrt{T_1}} = 1 \]

that is

    \[ M_{1}= M_{2} \]

Therefore, because the two similarity parameters (Reynolds number and Mach number) are the same for both flows, then the two flows will be dynamically similar.

Example #2

Consider two airfoils that have the same profile shape and operating angle of attack but have different chords and operating in two different fluids, as given in the table below. Determine whether or not the flows are dynamically similar.

Airfoil 1 Airfoil 2
\alpha_1 = 5^{\circ} \alpha_2 = 5^{\circ}
V_1 = 210 ms^{-1} V_2 = 140 ms^{-1}
\varrho_1 = 1.2 kg^{-3} \varrho_2 = 3.0 kg^{-3}
\mu_1 = 1.8 \times 10^{-5} kg m^{-1} s^{-1} \mu_2 = 1.5 \times 10^{-5} kg m^{-1} s^{-1}
a_1 = 300 ms^{-1} a_2 = 200 ms^{-1}
c_1 = 1.0 m c_2 = 0.5 m

The test for dynamic flow similarity requires us to determine the values of the similarity parameters, i.e., in this case the Reynolds number and Mach number for each flow. For Airfoil 1 we have for the Reynolds number

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

and for the corresponding Mach number

    \[ M_1 = \frac{V_1}{a_1} = \frac{210.0}{300.0} = 0.7 \]

For Airfoil 2 we have for the Reynolds number

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

and for the corresponding Mach number

    \[ M_2 = \frac{V_2}{a_2} = \frac{140}{200.0} = 0.7 \]

Therefore, despite the disparity in terms of the sizes of the airfoils and the different flow conditions, these two flows are indeed dynamically similar.

Similitude in Other Engineering Fields

As previously discussed, two systems can be considered geometrically, kinematically, and dynamically similar when all similarity parameters can be established to have the same numerical values, i.e., they are formally used to establish the scaling of the problem has similitude. In practice, however, complete similarity is difficult to achieve 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 in other fields besides aerodynamics. For example, scaling laws and similarity parameters can be derived in the field of structures, structural dynamics, aeroelasticity, hydrodynamics, etc. The ideas, again, are to scale the larger “real” or “full-scale” physical problem down to that of a model and/or a smaller prototype. If correctly done to match the similarity parameters governing the physics of the problem will then give the needed similarity of the physical behavior between the two disparate geometric scales.

However, just as in the field of aerodynamics, designing a scaled-down structure (e.g., a wing or complete flight vehicle) that can match all of the similarity parameters of the problem is very challenging. Nevertheless, relaxing one or more of the scaling parameters may be appropriate, allowing at least partial similitude to be obtained with a scaled model. To this end, parametric variations over some range of scales can sometimes expose the expected sensitivities of one parameter versus another and so emphasize the more important scaling parameter(s).

Structural & Structural Dynamic 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 often enforced. For most flight vehicles with relatively large deformations, inertial, gravitational, and restoring forces will all be important in the design of a sub-scale model.

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 the full-scale wing. Suppose the same material is used in both the model and prototype. In that case, it may not be possible to scale the properties of the model to obtain the same values of the similarity parameters, e.g., a 1:5 scale structural dynamic model would need to have five times the mass density to achieve appropriate scaling.

For structural dynamic similitude, the dimensionless parameters relevant to the scaled model are the reduced frequency, usually given the symbol k, and the Froude number, F\!r. Reduced frequencies are computed for a given airspeed considering the natural frequency of any given mode of vibration of the wing, \omega (in units of radians per second), and the wing span, b, i.e.,

    \[ k = \frac{\omega\, b}{V_{\infty}} \]

For a wing then the Froude number, F\!r, is normally defined as

    \[ F\!r = \frac{V_{\infty}}{\sqrt{ b \, g}} \]

which should be matched to ensure dynamic similarity when wing flexibility and aeroelastic effects are relevant.

The consideration of the time scales is also important, 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. However, for transient time response the reduced time, \hat{s}, may be used, i.e.,

    \[ \hat{s} = \frac{V_{\infty} \, t}{c} \]

where t is time and c is the wing chord. A physical interpretation of this non-dimensional time is the distance traveled by the wing through the flow in terms of chord lengths.

Aeroelastic Models

Aeroelastically-scaled models, which involve coupled aerodynamic-structure scaling, can also be designed to replicate their full-scale counterparts, such as for testing in the wind tunnel. However, the outcome usually results in a less stiff and more flexible model. The interaction between aerodynamics and the structural response appears in terms of the ratio between forces induced by dynamic pressure relative to the structure’s stiffness.

Suppose the actual airframe or wing is made of composite materials. In that case, structural scaling out of composites can be difficult to match such things as stiffnesses, natural frequencies, mode shapes, etc. Nevertheless, a major advantage of the 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, behavior that could surface during the first flights the full-scale vehicle.

An example of a wing flutter model. In this case, a proprotor replicating that used on a tiltrotor aircraft is mounted on a flexible, aeroelastically-scaled wing to study the potential for a whirl-flutter instability. This flutter effect is caused by the coupling of the proprotor aerodynamics and the proprotor’s overall dynamic response on the wing.

Hydrodynamic Models

Designing a new ship, yacht, seaplane, etc., 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, such as to improve its performance, i.e., by reducing the hydrodynamic drag on a ship as it moves through the water. The hydrodynamic drag on a ship hull is caused by both viscous effects (from viscous shear on the hull) as well as gravitational effects (from wave motion). In the latter case, a ship traveling over a sea leaves behind it a train of waves, and because these waves possess an energy that is eventually dissipated, then the ship experiences a wave drag force.

The hydrodynamic drag on the hull of a ship depends on the sum of skin friction (shear) drag and drag from wave motion.

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

    \[ C_D = f(R\!r, F\!r) \]

where R\!e is the Reynolds number based on ship length, l, and the non-dimensional grouping

    \[ F\!r= \frac{V_{\infty}}{\sqrt{g l}} \]

is the corresponding Froude number. The resistance from viscous effects is a function of Reynolds number and roughness of the hull. In model ship testing, separating these two components (sskin friction or shear drag and wave wake drag) makes it possible to determine the hydrodynamic drag of actual ships from tests done with smaller models, typically at 1:25 or 1:50 scale, such as in towing tanks.

The estimation of the hydrodynamic drag on a ship can be determined by testing a scaled model at or near the correct Reynolds number (which affects hull drag from viscous shear) and Froude number (which affects hull drag from wave motion).

It has been found that if the hydrodynamic Froude number is much less than unity, which implies that the wavelength of the waves is smaller than the ship’s length, then the resulting wave drag on the hull is comparatively small. If the ship’s length is equal to half the wavelength of the waves, the bow and stern waves interfere constructively, the so-called “hump speed” leading 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.

Speedboats, which can reach Froude numbers of over 3, can experience different regimes, eventually entering the so-called planing regime where they will skim over the water with significantly decreased drag. The same behavior occurs during the takeoff run of a seaplane when its weight eventually becomes supported by hydrodynamic lift rather than buoyancy, which is called operating “on the step.” The ability of the seaplane to successfully take off from the water and fly depends on the ability of the pilot to reach this operating condition.

Summary & Closure

The principles encompassing dynamic similarity are used in all branches and disciplines of engineering. While the development of appropriate similarity parameters can take some work, the benefits are usually significant. Similarity parameters not only help generalize the system’s behavior but also allow the engineer to reduce the scope of the problem, i.e., by reducing the number of dependent variables to consider, especially when there are many. similarity parameters facilitate the transfer of data and results from one scale to another.

The use of similarity parameters and the principles of dynamic similarity are essential tools in the design and optimization of engineering systems across many 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 more carefully studied, 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 be scaled to the actual conditions the aircraft will face in flight. This approach allows engineers to obtain valuable information and make predictions about the behavior of the aircraft without the need to conduct expensive, full-scale flight tests.

While the practice of using similarity parameters is by no means limited to wind tunnels, the testing of scaled models of aerospace systems, in general, can have much value in the process of engineering design. For example, the approach can help identify potential problems that can be designed out at an early stage rather than discovering them when the actual system is fielded.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

  • Discuss some of the challenges of using a gas other than air to achieve dynamic flow similarity with a scale model in a wind tunnel.
  • Do some research to determine the purpose of a cryogenic wind tunnel for testing scaled models.
  • What is  “flutter”? Do some research to determine why the potential for flutter can be a serious problem in airframe design.
  • Aeroelastic models are sometimes referred to as Froude-scaled models. What is the governing similarity parameter in this case?

Other Useful Online Resources about Dynamic Similarity

License

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

Introduction to Aerospace Flight Vehicles 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