7.2 Similarity parameters

Similarity parameters overview

Similarity parameters are dimensionless numbers that characterize how fluids behave and what forces act on objects within a flow. They capture the relative importance of different physical phenomena (viscosity, compressibility, gravity) so engineers can compare and scale different flow scenarios. This makes them essential for predicting the performance of aircraft, rockets, and other aerodynamic systems.

Importance in aerodynamics

Similarity parameters provide the framework for scaling wind tunnel experiments and computational simulations to full-scale aircraft. Without them, there's no reliable way to know whether results from a small model actually represent real-world conditions.

They also help identify critical flow regimes, like laminar vs. turbulent flow or subsonic vs. supersonic flow, which directly affect aerodynamic performance and design decisions.

Dimensionless numbers and ratios

A dimensionless number is formed by combining physical quantities so that all units cancel out, leaving a pure number. Each one represents the ratio of competing physical effects in a flow. For instance, the Reynolds number captures the ratio of inertial forces to viscous forces, while the Mach number captures the ratio of flow velocity to the speed of sound.

Because these ratios are unitless, they let you compare flow scenarios that look completely different on the surface. A small wind tunnel model and a full-size aircraft can share the same Reynolds number, which means the balance of forces is the same in both cases.

Reynolds number (Re)

The Reynolds number is the ratio of inertial forces to viscous forces in a fluid flow. It is defined as:

$Re = \frac{\rho V L}{\mu}$

where $\rho$ is fluid density, $V$ is flow velocity, $L$ is a characteristic length scale, and $\mu$ is dynamic viscosity.

Definition and formula

The formula combines fluid properties (density and viscosity) with flow characteristics (velocity and length scale) into a single dimensionless ratio. The characteristic length $L$ depends on the geometry: it could be the chord length of an airfoil, the diameter of a pipe, or the length of a flat plate.

You can also write it using kinematic viscosity $\nu = \mu / \rho$:

$Re = \frac{VL}{\nu}$

Physical meaning

The Reynolds number tells you which type of force controls the flow. Inertial forces tend to keep the fluid moving and amplify disturbances. Viscous forces tend to damp out disturbances and smooth the flow.

• High Re: Inertial forces dominate, and the flow tends to be turbulent and harder to predict.
• Low Re: Viscous forces dominate, and the flow tends to be laminar and stable.

Laminar vs. turbulent flow

• Laminar flow has smooth, parallel streamlines with minimal mixing between fluid layers.
• Turbulent flow has chaotic, swirling motions with enhanced mixing driven by inertial forces.
• The transition between the two depends on the Reynolds number and the geometry of the problem.

Critical Reynolds number

The critical Reynolds number is the value at which flow transitions from laminar to turbulent. For flow over a flat plate, this is approximately $5 \times 10^5$, though the exact value shifts depending on surface roughness and freestream turbulence levels. Knowing this transition point is essential for predicting boundary layer behavior and the onset of flow separation.

Applications in aerodynamics

• Characterizing flow around airfoils, wings, and fuselages
• Predicting boundary layer transition location, which affects skin friction drag and heat transfer
• Matching Reynolds number between wind tunnel models and full-scale aircraft to ensure test results are representative

Mach number (M)

The Mach number is the ratio of flow velocity to the local speed of sound:

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

where $V$ is the flow velocity and $a$ is the local speed of sound.

Definition and formula

The local speed of sound depends on fluid properties and temperature:

$a = \sqrt{\gamma R T}$

where $\gamma$ is the specific heat ratio (1.4 for air), $R$ is the specific gas constant, and $T$ is the absolute temperature. Because $a$ changes with temperature, the Mach number at a given airspeed varies with altitude and atmospheric conditions.

Physical meaning

The Mach number indicates how important compressibility effects are in a flow.

• Below about $M = 0.3$, density changes are negligible and the flow can be treated as incompressible.
• As $M$ increases beyond 0.3, compressibility effects grow, eventually producing shock waves and significant changes in fluid density, pressure, and temperature.

Subsonic vs. supersonic flow

• Subsonic: $M < 1$. Flow velocity is below the local speed of sound.
• Supersonic: $M > 1$. Flow velocity exceeds the local speed of sound.
• Transonic: The regime around $M \approx 0.8$ to $1.2$, where mixed subsonic and supersonic regions coexist on the same body.
• Hypersonic: Generally $M > 5$, where additional phenomena like aerodynamic heating become dominant.

Critical Mach number

The critical Mach number is the freestream Mach number at which the local flow first reaches $M = 1$ somewhere on the aerodynamic surface. For typical airfoils, this falls between 0.6 and 0.8, depending on the airfoil shape and angle of attack.

Exceeding the critical Mach number triggers shock wave formation and a sharp increase in drag called the transonic drag rise. This is a major design constraint for commercial transports cruising near $M = 0.85$.

Compressibility effects

As Mach number rises, compressibility produces several effects:

• Changes in fluid density, pressure, and temperature across the flow field
• Formation of shock waves, which are thin regions of abrupt property changes that increase drag and can cause flow separation
• Altered lift and drag characteristics that require compressible flow analysis methods (e.g., Prandtl-Glauert correction at lower transonic speeds)

Applications in aerodynamics

• Determining the appropriate flow regime (subsonic, transonic, supersonic, hypersonic) for design and analysis
• Designing high-speed aircraft, rockets, and missiles where compressibility is a primary concern
• Matching Mach number in wind tunnel tests and CFD simulations to capture compressibility effects accurately

Froude number (Fr)

The Froude number is the ratio of inertial forces to gravitational forces:

$Fr = \frac{V}{\sqrt{gL}}$

where $V$ is flow velocity, $g$ is gravitational acceleration, and $L$ is a characteristic length scale.

Definition and formula

The characteristic length $L$ depends on the problem: it could be the depth of a channel, the draft of a ship hull, or the height of a terrain feature. The Froude number tells you whether gravity or inertia controls the flow behavior.

Physical meaning

• High Fr: Inertial forces dominate, and the flow is less affected by gravity.
• Low Fr: Gravitational forces dominate, producing phenomena like surface waves and hydraulic jumps.
• The transition between gravity-dominated and inertia-dominated flow occurs around $Fr = 1$.

Applications in aerodynamics

The Froude number is most relevant in aerodynamic problems involving free surfaces or gravitational effects:

• Design of seaplanes and hydrofoils, where the interaction between air, water, and gravity matters
• Analysis of airflow over terrain, where gravity influences atmospheric wave formation
• Wind tunnel testing of ground vehicles, where ground proximity effects need proper scaling

Compared to Reynolds and Mach numbers, the Froude number comes up less often in pure aerodynamics, but it's critical in these specialized applications.

Strouhal number (St)

The Strouhal number relates oscillation frequency to flow velocity and a characteristic length:

$St = \frac{fL}{V}$

where $f$ is the oscillation frequency, $L$ is a characteristic length scale, and $V$ is the flow velocity.

Definition and formula

The characteristic length depends on geometry: the diameter of a cylinder, the chord of an airfoil, or the width of a bluff body. The Strouhal number quantifies how "unsteady" a flow is relative to its bulk motion.

Physical meaning

• High St: The flow is dominated by oscillatory behavior, with coherent vortical structures forming regularly.
• Low St: The flow is relatively steady, with minimal unsteady effects.

A classic example is vortex shedding behind a circular cylinder, where $St \approx 0.2$ over a wide range of Reynolds numbers. This predictable relationship is what makes the Strouhal number so useful.

Oscillating flows and vortex shedding

Vortex shedding occurs when alternating vortices detach from opposite sides of a bluff body, creating periodic pressure fluctuations. The Strouhal number characterizes the frequency and regularity of this shedding pattern. If the shedding frequency matches a structural natural frequency, dangerous resonance can occur (this is what contributed to the Tacoma Narrows Bridge collapse).

Applications in aerodynamics

• Predicting flow-induced vibrations of aircraft structures, antenna masts, and landing gear
• Analyzing unsteady loads on wings, control surfaces, and other components
• Matching Strouhal number in wind tunnel tests to ensure unsteady phenomena are properly scaled

Other similarity parameters

Beyond the four main parameters above, several other dimensionless numbers appear in aerodynamics and fluid mechanics, especially when heat transfer or rarefied gas effects are involved.

Prandtl number (Pr)

$Pr = \frac{\nu}{\alpha}$

The Prandtl number is the ratio of momentum diffusivity (kinematic viscosity $\nu$) to thermal diffusivity $\alpha$. It tells you whether momentum or heat diffuses faster through the fluid. For air at standard conditions, $Pr \approx 0.71$. This parameter is important whenever convective heat transfer matters, such as in high-speed flight where aerodynamic heating is significant.

Nusselt number (Nu)

$Nu = \frac{hL}{k}$

The Nusselt number is the ratio of convective to conductive heat transfer, where $h$ is the convective heat transfer coefficient, $L$ is a characteristic length, and $k$ is the fluid's thermal conductivity. A higher Nusselt number means convection is more effective at transferring heat. It's used in designing heat exchangers and thermal protection systems.

Grashof number (Gr)

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

The Grashof number is the ratio of buoyancy forces to viscous forces, where $\beta$ is the thermal expansion coefficient, $T_s$ and $T_\infty$ are surface and freestream temperatures, $L$ is a characteristic length, and $\nu$ is kinematic viscosity. It characterizes natural convection flows driven by temperature differences.

Knudsen number (Kn)

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

The Knudsen number is the ratio of the molecular mean free path $\lambda$ to a characteristic length $L$. It determines whether the continuum assumption of fluid mechanics holds:

• $Kn < 0.01$: Continuum flow (standard Navier-Stokes equations apply)
• $Kn > 0.1$: Rarefied flow (molecular effects become important)

This is critical for high-altitude aerodynamics (where air density is very low) and microfluidics applications.

Dynamic similarity

Dynamic similarity means that two geometrically similar flows have the same dimensionless force coefficients (lift coefficient, drag coefficient, pressure coefficient, etc.). When dynamic similarity holds, the balance of forces on the model is identical to the balance on the full-scale system.

Concept and importance

For wind tunnel results to apply to a real aircraft, the model flow must be dynamically similar to the full-scale flow. This means the relevant dimensionless parameters must match. If they do, you can directly scale force coefficients from the model to the real vehicle.

Without dynamic similarity, wind tunnel data can be misleading. A model might show attached flow where the real aircraft would have separation, or vice versa.

Maintaining similarity parameters

To achieve dynamic similarity, you need to match the relevant dimensionless parameters between model and full-scale conditions. This requires careful control of:

• Freestream velocity
• Fluid properties (density, viscosity, temperature)
• Model size

In practice, matching all parameters simultaneously is often impossible. For example, a 1/10th scale model at the same Mach number would need 10 times the Reynolds number per unit length to compensate for the smaller $L$. Engineers must decide which parameters matter most for the problem at hand.

Scaling and model testing

Common strategies for achieving or approximating dynamic similarity:

1. Pressurized wind tunnels increase air density $\rho$, which raises the Reynolds number without changing velocity or model size.
2. Cryogenic wind tunnels cool the air to reduce viscosity $\mu$, also raising the Reynolds number.
3. Heavy gas tunnels use gases denser than air (like $SF_6$) to modify fluid properties.
4. Partial similarity matches the most critical parameter (often Mach number for high-speed tests, or Reynolds number for low-speed tests) and corrects for the others analytically.

Scaling laws derived from dynamic similarity determine the appropriate model size, test velocity, and other conditions needed for accurate results.

Limitations of similarity parameters

Similarity parameters are powerful, but they rest on assumptions that don't always hold in practice.

Assumptions and simplifications

Each parameter is derived from a simplified model of fluid behavior. The Reynolds number, for instance, assumes Newtonian fluid behavior and constant fluid properties. These assumptions break down in certain situations, such as flows involving non-Newtonian fluids, chemically reacting gases (as in hypersonic re-entry), or large temperature gradients that cause fluid properties to vary significantly across the flow.

Real-world complexities

Real flows often involve complex geometries, unsteady behavior, and multiple interacting physical phenomena that a single dimensionless parameter can't fully capture. Flows with strong curvature, massive separation, or intense turbulent mixing may behave differently than what simple parameter matching would predict.

Combined effects of multiple parameters

Many real-world problems require matching several similarity parameters at once, which creates trade-offs. A classic example: high-speed, high-altitude flight requires matching both Reynolds number and Mach number. But in a conventional wind tunnel, achieving the correct Mach number on a small model typically results in a Reynolds number far below the full-scale value.

When you can't match everything, you need to prioritize. The general approach is:

• Match the parameter that most strongly governs the flow physics you're studying
• Use analytical corrections or empirical data to account for mismatches in other parameters
• Run tests at multiple conditions to bracket the real-world values and understand sensitivity

This judgment call is one of the most important skills in experimental aerodynamics.