Gravitational and Inertial Forces in Model Similarity

Gravitational and inertial forces are important forces that play a significant role in model similarity. When studying fluid flow, it is often necessary to create scaled-down models to conduct experiments in a laboratory setting. In order for these models to accurately represent real-world scenarios, the forces acting on the model must be properly scaled.

Discharge Ratio and Length Dimension

The discharge ratio, denoted as Lr, is a dimensionless parameter that relates the length dimension of the model to the original or prototype system. It is defined as the ratio of the length dimension of the model to the length dimension of the prototype. In this case, we are given that Lr is the ratio of length dimension.

Determining the Discharge Ratio

When gravitational and inertial forces are the only important forces in model similarity, we can determine the discharge ratio by considering the balance of these forces. Specifically, the discharge ratio is derived by equating the gravitational force and the inertial force.

Gravitational Force: The gravitational force acting on a fluid element is given by the product of the density of the fluid (ρ), the acceleration due to gravity (g), and the volume of the fluid element (V).

Inertial Force: The inertial force is related to the velocity (V) and the time derivative of the velocity (dV/dt) of the fluid element.

Equating the Gravitational and Inertial Forces

By equating the gravitational force and the inertial force, we obtain the following equation:

ρ * g * V = ρ * (dV/dt)

Here, ρ is the density of the fluid, g is the acceleration due to gravity, V is the velocity of the fluid element, and (dV/dt) is the time derivative of the velocity.

Simplifying the Equation

Simplifying the equation, we can cancel out the density term:

g * V = (dV/dt)

Integrating both sides of the equation, we get:

∫V dV = ∫g dt

Solving the integrals, we obtain:

(V^2) / 2 = g * t

Rearranging the equation, we find:

V = √(2 * g * t)

Applying the Discharge Ratio

Now, let's consider the length dimension. The length dimension (L) is related to the velocity (V) and the time (t) by the equation:

L = V * t

Substituting the value of V from the previous equation, we get:

L = (√(2 * g * t)) * t

Simplifying further, we find:

L = √(2 * g * t^3)

Since Lr is the ratio of the length dimension of the model to the length dimension of the prototype, we can write:

Lr = L_model / L_prototype

Substituting the equation for L, we have:

Lr = (√(2 * g * t_model^3)) / (√(2 * g * t_prototype^3))

Simplifying the equation, we find:

Lr = (√(t_model^3)) / (√(t_prototype^3))

Since t_model and t_prototype are constants, we can simplify further:

Lr = (t_model)^(3/2) /