I’ve often used similitude requirements to argue for the use of real-time hybrid simulation in fluid-structure interaction experiments. Wave basin and flume experiments are, often by necessity, scaled due to the size and capacity limitations of the facility. This often results in small-scale structural models. There are three types of similitude: geometric, kinematic, and dynamic.

- Geometric similitude is the similarity in shape between the model and prototype, defined by the length, $latex L$: $latex \lambda_L = L_p/L_m$
- Kinematic similitude is the similarity in the displacement, $latex u$, or velocity field, $latex v$, between the model and prototype: $latex \lambda_u = u_p/u_m$ or $latex \lambda_v = v_p/v_m$
- Dynamic similitude is the similarity between the governing dynamics (i.e., forces) between the model and prototype: $latex F_i+F_g+F_\mu+F_p+F_e+F_s+… = 0$, where $latex F_i =$ inertial forces, $latex F_g =$ gravitational forces, $latex F_\mu =$ viscous forces, $latex F_p =$ pressure forces, $latex F_e =$ elastic forces, $latex F_s =$ surface forces, and so on.

Perfect dynamic similitude requires that the dimensionless ratios between the forces (represented by Froude, Reynolds, Weber, Cauchy-Mach, Euler or Strouhal, among others) at prototype scale (full scale) are the same as those in the scaled model. However, in scaled experiments you can typically only scale one of the above forces at a time, $latex \lambda_F = F_p/F_m$. Since fulfilling perfect dynamic similitude for all of these dimensionless ratios is impossible, partial similitude is accepted and some forces are scaled while other are neglected.

Let’s consider the similitude of a structure subjected to a hurricane. For this case, we need to satisfy similitude for the structure, water (due to surge levels and waves), and wind. In this post, we’ll treat the domains separately (I’ll write future posts discussing each of these in more detail).

**Wind**: Wind can be defined by compressible flow, where the governing forces are the inertial and viscous forces ($latex F_i$ and $latex F_\mu$). Reynolds scaling ($latex Re$) is generally followed for wind experiments, where the governing forces are the inertia and viscous forces:

$latex Re = F_i/F_\mu = \rho v L/\mu$

For Reynolds scaling, the Reynolds number is the same at both the prototype and model scale:

$latex \lambda_{Re} = Re_p/Re_m = 1.0$

To meet this requirement, the time scale is $latex t_p/t_m = \lambda_L^2$, assuming the density ($latex \rho$) and viscosity ($latex \mu$) are the same in the prototype and model.

**Wave**: Waves can be defined by incompressible flow. Froude scaling ($latex Fr$) is generally followed for wave experiments, where the governing forces are inertia and gravity forces:

$latex Fr = F_i/F_g = v/\sqrt{gL}$

For Froude scaling, the Froude number is the same at both the prototype and model scale:

$latex \lambda_{Fr} = Fr_p/Fr_m = 1.0$

To meet this requirement, the time scale is $latex t_p/t_m = \sqrt{\lambda_L}$, assuming the gravitational constant ($latex g$) is the same in the prototype and model.

**Structure**: Structural dynamics depends on the structure’s stiffness, damping, and mass properties. In addition to force similitude, displacement similitude is often enforced. To satisfy similitude for the stress-strain constitutive relationship, Cauchy scaling ($latex Cy$) relates the inertia and restoring forces ($latex F_r$):

$latex Cy = F_i/F_r = \rho_s v_s^2/E$

For strongly nonlinear structures, inertial, gravitational, and restoring forces are all important, and both Froude and Cauchy scaling should be satisfied. However, if the same material is used in both the model and prototype, the mass in the model needs to scale by $latex 1/\lambda_L$, which is often too strong of a requirement for small-scale models.

Since specific weight scales the same as the length scale, it is often not possible to find a model material with the same strength properties that is $latex \lambda_L$ times denser than the prototype (e.g., if the model is built at 1:5 scale, it must have 5 times the mass of the prototype). It is common in earthquake engineering applications to artificially scale the mass while preserving density [i.e., $latex (g\rho_s L/E)_m=(g\rho_s L/E)_p$] to achieve acceleration similitude in the model and prototype. In this case, the time scale is $latex t_p/t_m = \sqrt{\lambda_L}$ and is equivalent to Froude scaling.

In contrast, hydro- and aero-dynamic experiments often neglect structural similitude (e.g., in terms of stiffness, mass, etc.). In the case where the gravity forces (scaling of specific weight $latex \gamma$) are neglected (i.e., pure Cauchy similitude), the time scale is $latex t_p/t_m = \lambda_L$, assuming the modulus of elasticity ($latex E$), Poisson’s ratio ($latex \nu$), and the mass density ($latex \rho_s$) are the same in the prototype and model and neglecting scaling of $latex g$.

Any scaling in time also applies to the structure’s period, causing the model to have higher frequencies than the prototype.

Note, I’m approaching this from an earthquake engineering perspective. Fluid-elastic similitude with aero- and hydro-elastic models have been documented extensively by others.

**Coupling**: To couple everything together (wind, wave, and structure) the time scale needs to be consistent across the domains. For example, the figure below shows the variation in the ratio of the model-to-prototype time scale, $latex t_m/t_p$, depending on Froude versus Reynolds similitude. The time scale is the same only at full scale ($latex L_m/L_p = 1.0$). A small scale model results in incompatibility in the time scale between the wind and waves. For fluid-structure interaction, scaling of the interaction between the fluid and structure (e.g., by the mass $latex M = \rho/\rho_s$, reduced velocity $latex Vr = v/\sqrt{E/\rho_s}$, or Cauchy number in terms of the ratio between dynamic pressure from the fluid to the stiffness of the solid) could also be important.

Selection of the scaling laws depends on what you consider to be most important to the problem. If you have a problem involving multiple physical laws, the simulation can be subject to incompatibilities in the scaling laws. Choosing what to scale depends on the the magnitude of one dimensionless number with respect to the others. If a single dimensionless number is large, you may be in luck and can neglect many of the other forces. However, for multi-physics problems where multiple dimensionless numbers are important (i.e., like fluid-structure interaction problems), it may not be possible to select only one scale law.

Hybrid simulation couples physical experiments to numerical models. Separately, the physical and numerical sub-assemblies represent parts of the complete assembly. Combined, the hybridized physical-numerical sub-assemblies can represent full assembly response. This can be advantageous when considering similitude requirements for multi-physics problems. Some scaling effects can be mitigated by housing some of the full-scale prototype properties in the numerical model. While it may not be possible to model every aspect of the problem with perfect similitude, hybrid simulation may allow you to get closer to the full-scale prototype response.

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