Problem solving procedures

The Buckingham Pi procedure

Step 1: count the variables

List every variable involved in the system. Count them to find your total number of variables ($n$).

Step 2: determine the dimensions

Write down the primary dimensions for each variable using brackets, like $\{M L T^{-2}\}$ for force. Count how many unique primary dimensions appear across your entire list. This number is $j$.

Step 3: calculate the number of Pi groups

Subtract your dimensions from your variables ($k = n - j$). This is the exact number of dimensionless groups you need to find.

Step 4: pick your repeating variables

You need to pick a few variables to be your "repeaters." The number of repeaters must exactly equal your number of dimensions ($j$). These variables will act as the foundation for every Pi group you build. However, if you pick the wrong ones, the math will completely fail. You must obey these three strict rules:

• Rule 1: They must cover every dimension. If your overall problem uses Mass, Length, and Time, your chosen repeaters must collectively contain Mass, Length, and Time. You cannot pick three variables that only use Length and Time.
• Rule 2: They cannot cancel each other out. Your repeaters cannot combine to form a dimensionless group on their own. For example, you cannot pick Length ($L$), Time ($T$), and Velocity ($V$) together. Why? Because Velocity is just Length divided by Time. They would cancel each other out and break the algebra.
• Rule 3: No two variables can have the same dimensions. This is an absolute rule. If your problem involves both a pipe length ($L$) and a pipe diameter ($D$), you cannot pick both as repeaters because they both have the exact same dimension: Length $\{L\}$. You must choose just one.

Step 5: use the shortcut for duplicate dimensions

Before doing any math, look at your leftover (non-repeating) variables. If any non-repeating variable has the exact same dimension as one of your repeaters, you get a free Pi group. Just divide the non-repeater by the repeater.

Step 6: set up the algebraic balance

For the remaining variables, multiply your repeaters by exactly one non-repeating variable. Give the repeaters unknown exponents ($a, b, c$). Set the whole product equal to a perfectly dimensionless state: $\{M^0 L^0 T^0\}$.

Step 7: solve for the exponents

Substitute the actual primary dimensions into the brackets. Create a simple equation for each dimension (like setting all the Mass exponents equal to zero). Solve these equations to find the numerical values for your exponents.

Step 8: assemble and repeat

Apply the solved exponents to the original variables to finish the Pi group. A negative exponent moves that variable to the bottom of the fraction. Repeat steps 6 through 8 for every leftover variable until you find all $k$ groups.

Equation scaling (star variable approach)

Step 1: find physical variables and constants

First, identify the physical variables that are changing in the equation, like local velocity ($u$) or time ($t$). Then, identify the constant reference scales the problem gives you, like average velocity ($U_0$) or total length ($L$).

Step 2: build the star variables

Divide the physical variable by a constant that has the exact same dimension. This cancels the units and creates your dimensionless star variable. For example: $u^* = u / U_0$.

Step 3: isolate the physical variables

Rearrange your star variable equations to isolate the physical variable. Move the constant to the other side to create a replacement equation: $u = u^* U_0$.

Step 4: substitute into the equation

Replace every physical variable in the original equation with your new replacement packages.

Step 5: manage the calculus operators

It is critical to understand how dimensions behave inside calculus operators. A derivative is simply a ratio of small changes, so the dimensions divide. An integral is a sum of small products, so the dimensions multiply. The operators themselves do not magically add or change dimensions.

Because your reference scales are constants, standard calculus rules allow you to pull them completely outside of the derivative or integral. Group these constants cleanly at the front of each term.

Step 6: normalize the equation

Your equation is now full of dimensionless derivatives multiplied by groups of constants. Divide the entire equation by the group of constants attached to your main driving term.

Step 7: identify the final groups

This division strips all physical units from the equation. The remaining chunks of constants will naturally form standard dimensionless numbers, like the Reynolds or Froude numbers.

Setting up the energy equilibrium

Step 1: pick your two points

To build your equation, you first need to choose a starting point and an ending point. The absolute best points to pick are the extremities of the system, like the top free surface of a large reservoir or the open exit of a pipe discharging straight into the atmosphere.

Step 2: build and simplify the balance equation

The total energy at your starting point, plus any energy added by a pump, minus any energy extracted by a turbine, minus all the energy lost to friction and components, must equal the total energy at your ending point.

Once you write this out, the second step is to immediately simplify it. Cross out everything that is not straightforward. If a term is zero, delete it. If you have the exact same term on both sides of the equation, cancel them out.

Step 3: mind your units

You can write this energy equation in terms of height, meaning meters, or you can write it in terms of pressure. They are exactly the same thing, and you can swap between them just by multiplying or dividing by fluid density and gravity ($\rho g$).

When you express the losses in terms of height, you need to multiply your loss coefficients by $\frac{V^2}{2g}$. If you express them in terms of pressure, you multiply them by $\frac{1}{2}\rho V^2$. Usually, if you have a reservoir or two points at different elevations, writing it in terms of height is a lot more straightforward.

Calculating the flow losses

Step 1: check the internal Reynolds number limits

Before you can calculate pipe friction, you must know if the flow is laminar or turbulent. For internal pipe flow, the magic numbers are 2,000 and 4,000. If the Reynolds number is below 2,000, the flow is laminar. If it is above 4,000, it is turbulent.

Step 2: find the distributed major losses

Distributed losses happen because the fluid is constantly rubbing against the long walls of the pipe. You will usually find the friction factor ($f$) using the Haaland equation based on the pipe material.

Step 3: add up the minor losses

Minor losses are the ones that occur at single, specific points along your system rather than across a long length. Every time the pipe turns a bend, enters a tank, exits a tank, or passes through something like a filter, you lose energy. You will need a specific loss coefficient for each of these features.

Flat plate boundary layers

Step 1: check the external Reynolds limit

The limits for external flow are totally different from internal pipes. For a flat plate, the flow is much more stable, and we assume transition happens when the Reynolds number reaches exactly 500,000 ($5 \times 10^5$). Anything below that is laminar, and anything above that is turbulent.

Step 2: test the trailing edge

Because the Reynolds number grows with distance, the biggest number will always be at the very end of the plate. Start by calculating the Reynolds number at this trailing edge.

If that final number is smaller than 500,000, it means the flow never had a chance to transition. The entire boundary layer is laminar throughout the surface, and you can automatically apply the laminar equations.

If the number is bigger than 500,000, the flow became turbulent somewhere along the plate. You need to find exactly where that point is by setting the Reynolds number to 500,000 and solving for the distance ($x$).

If the transition happens after the halfway point, we just assume the whole boundary layer is laminar to keep things simple. If it happens before the halfway mark, we assume it is mostly turbulent and use the turbulent equations for the whole thing.

Step 3: calculate the surface area correctly

When you calculate the final drag force, you need to know the total area touching the fluid. The length is the dimension extending along the flow, and the width is usually perpendicular, going into the screen.

Drag on bulky 3D objects

Step 1: skip the flat plate equations

If you are dealing with an object like a sphere, a cube, or a car, we do not use the complex equations we just discussed for flat plates.

Step 2: read the drag coefficient from a table

For these shapes, the Drag Coefficient ($C_D$) has already been mapped out through testing. You will just need to look up the specific object in a table, find its coefficient, and plug it directly into the standard drag force formula.