Dimensional analysis and similarity

A consolidated reference of the fundamental dimensional rules, scaling formulations, PDE normalization variables, and kinematic relations utilized across the dimensional analysis and similarity problem set.

Principle in use: Dimensional homogeneity. Every valid physical equation must balance dimensionally. Before extracting complex non-dimensional groups, we must be able to isolate and identify the primary $MLT\Theta$ dimensions of composite expressions, including those containing calculus operators.

Problem statement: Report the primary dimensions $\{M L T \Theta\}$ for each composite expression listed below:

(a) $\rho u \frac{\partial u}{\partial x}$
(b) $\int_1^2 (p - p_0) dA$
(c) $\rho c_p \frac{\partial^2 T}{\partial x \partial y}$
(d) $\iiint \rho \frac{\partial u}{\partial t} dx dy dz$

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass ($M$), Length ($L$), Time ($T$), and Temperature ($\Theta$) continuum basis. Symbols denote measurable macroscopic fields.

Part (a): convective acceleration term $\left(\rho u \frac{\partial u}{\partial x}\right)$

First, identify the primary dimensions of each individual variable:

• Density $\rho = \{M L^{-3}\}$
• Velocity $u = \{L T^{-1}\}$
• Spatial derivative $\frac{\partial u}{\partial x} = \frac{\{L T^{-1}\}}{\{L\}} = \{T^{-1}\}$

Multiply the individual dimensions together to find the composite dimension:

Physical intuition: This term appears in the Navier-Stokes momentum equation. It represents a force per unit volume. We can verify this: Force is $\{M L T^{-2}\}$, and dividing by volume $\{L^3\}$ yields $\{M L^{-2} T^{-2}\}$.

Part (b): pressure area integral $\left(\int_1^2 (p - p_0) dA\right)$

This expression contains a difference and an integral.

• The difference: Subtracting a reference pressure $p_0$ from a local pressure $p$ simply yields a relative pressure difference. By the rule of homogeneity, you can only subtract terms with identical dimensions. Thus, $(p - p_0)$ retains the standard dimension of pressure: $\{M L^{-1} T^{-2}\}$.
• The integral: The integral sign and the numerical bounds ($1$ and $2$) are dimensionless indices. The differential $dA$ carries the dimension of Area: $\{L^2\}$.

Multiply the pressure dimension by the area dimension:

Physical intuition: The resulting dimension $\{M L T^{-2}\}$ is the exact primary dimension for Force. This makes physical sense: integrating a pressure distribution across a solid surface yields the net aerodynamic or hydrostatic force.

Part (c): mixed second derivative $\left(\rho c_p \frac{\partial^2 T}{\partial x \partial y}\right)$

Identify the dimensions of specific heat $c_p$ and the spatial derivative:

• Density $\rho = \{M L^{-3}\}$
• Specific heat $c_p$ represents energy per unit mass per degree Kelvin ($\text{J/(kg}\cdot\text{K)}$). Since Energy is Work (Force $\times$ Distance = $\{M L^2 T^{-2}\}$), we evaluate: $c_p = \frac{\{M L^2 T^{-2}\}}{\{M\} \{\Theta\}} = \{L^2 T^{-2} \Theta^{-1}\}$.
• Derivative $\frac{\partial^2 T}{\partial x \partial y} = \frac{\{\Theta\}}{\{L\}\{L\}} = \{\Theta L^{-2}\}$

Multiply the three terms together:

Part (d): volume integral of local acceleration $\left(\iiint \rho \frac{\partial u}{\partial t} dx dy dz\right)$

Identify the dimensions of the local temporal acceleration and the volume differential:

• Density $\rho = \{M L^{-3}\}$
• Local acceleration $\frac{\partial u}{\partial t} = \frac{\{L T^{-1}\}}{\{T\}} = \{L T^{-2}\}$
• Volume differential. Integrating across three spatial dimensions ($dx$, $dy$, $dz$) multiplies by Length three times: $\{L\} \times \{L\} \times \{L\} = \{L^3\}$.

Multiply the terms together:

Physical intuition: Notice that density $\{M L^{-3}\}$ multiplied by a volume differential $\{L^3\}$ yields Mass $\{M\}$. We are essentially multiplying Mass by Acceleration $\{L T^{-2}\}$. By Newton's Second Law, this correctly yields the dimension of Force.

Principle in use: Dimensional homogeneity and algebraic exponent matching. By enforcing that a derived parameter must be perfectly dimensionless ($\{M^0 L^0 T^0\}$), we can systematically solve for the unknown exponents of its constituent variables to uncover the governing physical ratio.

Problem statement: Form a dimensionless Stokes number ($St$) from the acceleration of gravity ($g$), dynamic viscosity ($\mu$), fluid density ($\rho$), characteristic particle speed ($U$), and characteristic diameter ($D$).

(a) If $St$ is directly proportional to $\mu$ and inversely proportional to $g$, determine its exact algebraic formulation.
(b) Demonstrate that $St$ is mathematically equivalent to the quotient of two traditional dimensionless groups.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Part (a): exact formulation of the Stokes number

First, establish the primary dimensions for the five specified variables:

• Gravity $g = \{L T^{-2}\}$
• Dynamic viscosity $\mu = \{M L^{-1} T^{-1}\}$
• Density $\rho = \{M L^{-3}\}$
• Velocity $U = \{L T^{-1}\}$
• Diameter $D = \{L\}$

Construct the baseline equation using the given anchors and the unknown exponents ($a, b, c$):

Substitute the primary dimensions into the equation and set the product equal to $\{M^0 L^0 T^0\}$:

Distribute the exponents across the terms to group them by primary dimension:

Extract a linear equation for each primary dimension by adding the respective exponents together and setting the sum to zero. Solve them in order of simplicity:

• Mass ($M$): $1 + a = 0 \implies \mathbf{a = -1}$
• Time ($T$): $-1 + 2 - b = 0 \implies 1 - b = 0 \implies \mathbf{b = 1}$
• Length ($L$): $-1 - 1 - 3a + b + c = 0$
  Substitute the knowns ($a = -1$, $b = 1$):
  $-2 - 3(-1) + 1 + c = 0$
  $-2 + 3 + 1 + c = 0 \implies 2 + c = 0 \implies \mathbf{c = -2}$

Substitute the solved exponents back into the baseline formula. Negative exponents move the variable to the denominator:

Part (b): resolving into traditional groups

The task is to express this specific derived result as a quotient of two universally recognized dimensionless groups.

Multiply the numerator and denominator of the Stokes formulation by $U$:

Rearrange the variables into two distinct fractional clusters:

Substitute the defined traditional groups back into the arrangement:

Principle in use: Dimensional exponent matching (the Rayleigh method). By proposing a generalized power-law relationship between a target variable and its governing physical parameters, we can solve for the unknown exponents to derive a strictly homologous scaling law.

Problem statement: For deep-water capillary waves, physical observation indicates that the phase speed $C$ depends exclusively on the fluid density $\rho$, the wavelength $\lambda$, and the surface tension $\Upsilon$.

Derive the analytical scaling law for the phase speed $C$, up to a dimensionless constant. Subsequently, determine the exact percent change in $C$ when the surface tension $\Upsilon$ is doubled, assuming $\rho$ and $\lambda$ remain fixed.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We evaluate the system utilizing the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Part (a): dimensional derivation of the scaling law

First, establish the primary dimensions for the target variable and the independent variables:

• Wave speed $C = \{L T^{-1}\}$
• Density $\rho = \{M L^{-3}\}$
• Wavelength $\lambda = \{L\}$
• Surface tension $\Upsilon = \{M T^{-2}\}$ (Derived from Force per unit length: $\{M L T^{-2}\} / \{L\}$)

Construct the functional equation using the unknown exponents and the dimensionless constant:

Substitute the primary dimensions into both sides of the equation. Because the wave speed ($C$) lacks a Mass dimension, we explicitly write it as $M^0$ to maintain alignment for the algebraic extraction:

Distribute the exponents across the bracketed terms and group the matching dimensions on the right side by adding their exponents:

Extract a linear algebraic equation for each primary dimension and solve for the unknown exponents:

• Mass ($M$): $0 = a + c \implies a = -c$
• Time ($T$): $-1 = -2c \implies \mathbf{c = \frac{1}{2}}$
  Substituting $c$ back into the Mass equation yields $\mathbf{a = -\frac{1}{2}}$.
• Length ($L$): $1 = -3a + b$
  Substitute $a$: $1 = -3\left(-\frac{1}{2}\right) + b \implies 1 = \frac{3}{2} + b \implies \mathbf{b = -\frac{1}{2}}$

Substitute the calculated exponents back into the initial functional equation to reveal the finalized scaling law:

Part (b): evaluating the parametric change

We are asked to evaluate how the wave speed changes if the surface tension is doubled (from $\Upsilon$ to $2\Upsilon$), while the fluid density ($\rho$) and the wavelength ($\lambda$) remain constant. We define this new phase speed as $C_{\text{new}}$.

Substitute $2\Upsilon$ into our derived scaling law:

We can factor the constant multiplier $2$ out of the square root operator. The remaining grouped terms are mathematically identical to our original baseline equation for $C$:

The velocity increases by a factor of $\sqrt{2}$, which is approximately $1.414$. To determine the strict percentage increase, we subtract the original baseline ($1.0$), matching the standard textbook reporting convention:

Principle in use: The Buckingham Pi Theorem. By enforcing dimensional homogeneity across a system of physical variables, we systematically reduce the number of independent parameters required to characterize a complex fluid phenomenon, forming universal dimensionless scaling groups.

Problem statement: Model the wall shear stress ($\tau_{\text{w}}$) as a dependent function of the free-stream velocity ($U$), boundary layer thickness ($\delta$), characteristic turbulence velocity fluctuation ($u'$), fluid density ($\rho$), and the axial pressure gradient ($dp/dx$) within a developing boundary layer.

Apply the Buckingham Pi theorem utilizing the repeating variable set ($\rho$, $U$, $\delta$) to express the physical system as a dimensionless functional relationship.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Step 1: system definition and the dimensional matrix

First, we list every physical variable involved and determine its primary dimensions.

• Wall shear stress $\tau_{\text{w}} = \{M L^{-1} T^{-2}\}$
• Stream velocity $U = \{L T^{-1}\}$
• Boundary layer thickness $\delta = \{L\}$
• Turbulence velocity fluctuation $u' = \{L T^{-1}\}$
• Fluid density $\rho = \{M L^{-3}\}$
• Pressure gradient $dp/dx = \{M L^{-2} T^{-2}\}$

Step 2: deriving the Pi groups

To derive each group, we force the product of the dimensional variables to equal a perfectly dimensionless state, mathematically defined as $\{M^0 L^0 T^0\}$.

Deriving $\Pi_1$ (targeting $\tau_{\text{w}}$):

• Construct the balance: $\Pi_1 = \rho^a U^b \delta^c \tau_{\text{w}}^1$
• Substitute dimensions: $\{M^0 L^0 T^0\} = \{M L^{-3}\}^a \{L T^{-1}\}^b \{L\}^c \{M L^{-1} T^{-2}\}$
• Solve for Mass ($M$): $a + 1 = 0 \implies \mathbf{a = -1}$
• Solve for Time ($T$): $-b - 2 = 0 \implies \mathbf{b = -2}$
• Solve for Length ($L$): $-3a + b + c - 1 = 0 \implies -3(-1) - 2 + c - 1 = 0 \implies \mathbf{c = 0}$
• Substitute back into the balance: $\Pi_1 = \rho^{-1} U^{-2} \delta^0 \tau_{\text{w}}^1 = \mathbf{\frac{\tau_{\text{w}}}{\rho U^2}}$

Deriving $\Pi_2$ (targeting $u'$):

• Applying the inspection shortcut directly yields: $\Pi_2 = \mathbf{\frac{u'}{U}}$

Deriving $\Pi_3$ (targeting $dp/dx$):

• Construct the balance: $\Pi_3 = \rho^a U^b \delta^c (dp/dx)^1$
• Substitute dimensions: $\{M^0 L^0 T^0\} = \{M L^{-3}\}^a \{L T^{-1}\}^b \{L\}^c \{M L^{-2} T^{-2}\}$
• Solve for Mass ($M$): $a + 1 = 0 \implies \mathbf{a = -1}$
• Solve for Time ($T$): $-b - 2 = 0 \implies \mathbf{b = -2}$
• Solve for Length ($L$): $-3a + b + c - 2 = 0 \implies -3(-1) - 2 + c - 2 = 0 \implies 3 - 4 + c = 0 \implies \mathbf{c = 1}$
• Substitute back into the balance: $\Pi_3 = \rho^{-1} U^{-2} \delta^1 (dp/dx)^1 = \mathbf{\frac{\delta}{\rho U^2} \frac{dp}{dx}}$

Step 3: final functional form and physical interpretation

The Buckingham Pi theorem concludes by stating that the dependent dimensionless group ($\Pi_1$) can be written as an arbitrary function ($fcn$) of the remaining independent dimensionless groups ($\Pi_2$ and $\Pi_3$).

Principle in use: Dimensional exponent matching for thermodynamic systems. By enforcing dimensional homogeneity across thermal and kinematic variables, we systematically derive the Stanton number—a fundamental dimensionless group used to characterize convective heat transfer.

Problem statement: Convective heat transfer data is typically parameterized using a convective heat transfer coefficient ($h$), defined by Newton's Law of Cooling: $\dot{Q} = h A \Delta T$. In this relation, $\dot{Q}$ is the absolute heat flow rate ($\text{J/s}$ or Watts), $A$ is the wetted surface area ($\text{m}^2$), and $\Delta T$ is the macroscopic temperature difference ($\text{K}$) between the solid surface and the fluid.

The Stanton number ($St$) is a dimensionless parameter built exclusively from $h$, fluid density ($\rho$), specific heat capacity ($c_p$), and the free-stream velocity ($V$).

First, determine the primary dimensions of $h$. Then, derive the exact algebraic formulation for $St$ under the strict assumption that $St$ scales linearly with $h$.

Note: The Stanton number ($St$) is entirely distinct from the Stokes number and Strouhal number, despite frequently sharing the same abbreviation in fluid literature.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time-Temperature ($MLT\Theta$) continuum basis to establish primary dimensions.

Step 1: defining the dimensions of the heat transfer coefficient ($h$)

First, algebraically rearrange Newton's Law of Cooling to isolate the target variable, $h$. Then, establish the primary dimensions for the remaining known variables:

• Heat flow rate $\dot{Q} = \{M L^2 T^{-3}\}$
• Surface area $A = \{L^2\}$
• Temperature difference $\Delta T = \{\Theta\}$

Substitute these primary dimensions into the rearranged equation to extract the dimensional signature of $h$:

Step 2: deriving the Stanton number formulation

The problem specifies that the Stanton Number is directly proportional to $h$, dictating that $h$ must carry an exponent of exactly $1$. We assign unknown algebraic exponents ($b, c, d$) to the remaining independent variables ($\rho, c_p, V$):

Establish the primary dimensions for these remaining variables. For specific heat ($c_p$), we recall its definition: Energy required per unit mass per unit temperature rise:

• Density $\rho = \{M L^{-3}\}$
• Specific heat $c_p = \frac{\text{Energy}}{\text{Mass} \cdot \text{Temperature}} = \frac{\{M L^2 T^{-2}\}}{\{M\}\{\Theta\}} = \mathbf{\{L^2 T^{-2} \Theta^{-1}\}}$
• Velocity $V = \{L T^{-1}\}$

Substitute the dimensions into the formulation and equate the product to a perfectly dimensionless thermodynamic state $\{M^0 L^0 T^0 \Theta^0\}$:

Extract the linear equations for Mass, Temperature, and Time, and solve the system sequentially:

• Mass ($M$): $1 + b = 0 \implies \mathbf{b = -1}$
• Temperature ($\Theta$): $-1 - c = 0 \implies \mathbf{c = -1}$
• Time ($T$): $-3 - 2c - d = 0 \implies -3 - 2(-1) - d = 0 \implies -1 - d = 0 \implies \mathbf{d = -1}$

Now, utilize the final dimension (Length) to verify the mathematical integrity of the solution:

• Length ($L$) verification: $-3b + 2c + d = -3(-1) + 2(-1) - 1 = 3 - 2 - 1 = \mathbf{0}$. The dimensional balance is perfectly verified.

Substitute the exponents back into the proportionality equation. Because all unknown exponents resolved to $-1$, they are moved cleanly into the denominator:

Principle in use: Equation-based scaling. By systematically substituting reference scales into a governing partial differential equation (PDE) and normalizing the operators, the inherent dimensionless parameters ($\Pi$ groups) naturally emerge as algebraic coefficients.

Problem statement: Incompressible 2-D flow in a Darcy-type porous solid obeys the following approximate thermodynamic energy relation:

Here $\sigma$ denotes permeability; all remaining symbols retain their standard thermofluid definitions.

(a) Determine the appropriate primary dimensions for permeability ($\sigma$).
(b) Nondimensionalize this equation, utilizing $(L, U, \rho, T_0)$ strictly as the reference scaling constants, and discuss the physical meaning of the dimensionless parameter that surfaces.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time-Temperature ($MLT\Theta$) continuum basis to establish primary dimensions.

Part (a): dimensional extraction via homogeneity

To determine the unknown dimensions of permeability ($\sigma$), we must algebraically equate a term containing $\sigma$ with a term that does not.

First, evaluate the known dimensions for the third term, $k \frac{\partial^2 T}{\partial y^2}$. Recall that thermal conductivity ($k$) is defined dimensionally as $\{M L T^{-3} \Theta^{-1}\}$, and a second-order spatial derivative divides the operand by Length squared:

Next, establish the known dimensions of the first term (temporarily excluding $\sigma$): $\rho c_p \frac{1}{\mu} \frac{\partial p}{\partial x} \frac{\partial T}{\partial x}$.

Consolidate the primary dimensions by aggregating their respective exponents:

By the rule of dimensional homogeneity, the entire first term (which includes $\sigma$) must perfectly equal the established dimensions of the third term:

Divide to isolate and resolve $\{\sigma\}$:

Physical intuition: The dimension of permeability ($\sigma$) is strictly an Area. This represents the effective cross-sectional "flow area" available within the porous rock or filter matrix.

Part (b): nondimensionalizing the PDE

Step 1: define the dimensionless (starred) variables. The prompt dictates using $(L, U, \rho, T_0)$ as the scaling foundation. To scale pressure ($p$) utilizing only these constants, we construct a dynamic pressure scale ($\rho U^2$):

Step 2: expand the physical differentials. Express the original derivatives utilizing the starred variables and scaling factors:

Step 3: substitute into the PDE. Replace the operators in the original equation. Note that the $x$-derivative and $y$-derivative terms will share the exact same scaling constants:

Step 4: normalize the equation. Divide the entire equation by the dimensional coefficient of the second-derivative term ($k T_0 / L^2$) to cleanly isolate it:

The Length ($L^2$) and Temperature ($T_0$) scales cancel perfectly. The remaining cluster of variables is completely dimensionless. We define this single parameter as $\zeta$:

Principle in use: Equation-based scaling for transport phenomena. By applying the Principle of Dimensional Homogeneity, we extract the primary dimensions of unknown transport coefficients. Subsequently, by normalizing the governing partial differential equation (PDE), we mathematically derive the fundamental dimensionless groups (Péclet and Damköhler numbers) that control species concentration.

Problem statement: Species transport in a plug-flow reactor is represented by the following reaction-advection-diffusion equation:

Here $u$ is the streamwise velocity, $\mathcal{D}$ is a diffusion coefficient, $k$ is a reaction-rate constant, $x$ measures distance along the reactor, and $C$ is a dimensionless species concentration.

(a) Determine the appropriate primary dimensions of $\mathcal{D}$ and $k$.
(b) Using a characteristic length scale $L$ and average velocity $V$ as parameters, rewrite this equation in dimensionless form and comment on any Pi groups appearing.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Part (a): dimensional extraction via homogeneity

Because all terms contain $C$, we can securely establish the unknown dimensions of $k$ and $\mathcal{D}$ by comparing their respective terms to the known advective term $\left\{ u \frac{\partial C}{\partial x} \right\}$:

Equate the reaction rate term ($kC$) to this required dimension:

Next, equate the diffusion term $\left( \mathcal{D} \frac{\partial^2 C}{\partial x^2} \right)$ to the required dimension, recalling that a second-order spatial derivative divides the operand by Length squared ($\{L^2\}$):

Part (b): nondimensionalizing the PDE

Step 1: define the dimensionless (starred) variables. The concentration $C$ is already dimensionless, so it does not require a starred transformation ($C^* = C$).

Step 2: expand the physical differentials. Express the original physical differentials strictly in terms of the starred differentials and their scaling constants:

Step 3: substitute into the PDE. Replace the physical variables and operators in the basic partial differential equation with their scaled equivalents:

Pull the scaling constants out to the front of each term to clearly isolate the operational coefficients:

Step 4: normalize the equation. To strip away the dimensional units, divide every term in the equation by the dimensional coefficient of the advective term ($V/L$). This forces the advective multiplier to equal exactly $1$:

Principle in use: Dynamic similarity and steady-state mechanical equilibrium. By establishing that the dimensionless drag coefficient ($C_{\text{D}}$) for a bluff body is independent of the fluid medium at high Reynolds numbers, we can transfer wind-tunnel data (simulating Mars) directly to an Earth-atmosphere force balance to estimate the supported payload.

Problem statement: A parachute sized for Mars descent was characterized in a wind tunnel with a drag coefficient (see Reference table) of $C_{\text{D}} \approx 1.1$ based on its projected frontal area.

Re-evaluate the load capacity for an Earth flight: at a $1000\text{ m}$ altitude, the canopy falls at a steady terminal speed of $18\text{ mi/h}$. Estimate the total supported weight.

Given operational data:
Descent speed $V = 18\text{ mi/h}$
Parachute diameter $D = 51\text{ ft}$

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Step 1: dimensional standardization

Execute the exact conversions for the operational velocity and parachute geometry:

• Velocity ($V$): $18\text{ mi/h} \times \left(\frac{0.44704\text{ m/s}}{1\text{ mi/h}}\right) \approx \mathbf{8.047\text{ m/s}}$
• Diameter ($D$): $51\text{ ft} \times 0.3048 = \mathbf{15.545\text{ m}}$

Calculate the projected frontal reference area ($A$) of the fully inflated canopy:

Step 2: physical principles of steady flight

Step 3: calculating the payload capacity

Substitute the rigorously standardized SI values into the established force balance equation to isolate the total weight ($W$):

Principle in use: Dynamic similarity and empirical drag modeling. By calculating the characteristic Reynolds number, we can extract dimensionless drag and shedding coefficients from established experimental data to compute macroscopic towing power and wake frequencies.

Problem statement: A towed sonar body is modeled as a circular cylinder $1\text{ ft}$ in diameter and $30\text{ ft}$ long, with its axis oriented strictly perpendicular to the tow direction. At a constant cruising speed of $12\text{ knots}$ ($1\text{ kn} = 1.69\text{ ft/s}$), estimate the active towing power required (in horsepower) and the vortex-shedding frequency generated in the wake.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Step 1: dimensional standardization and kinematic state

Execute the exact conversions for the geometry and velocity:

• Diameter ($D$): $1\text{ ft} \times 0.3048 = \mathbf{0.3048\text{ m}}$
• Length ($L$): $30\text{ ft} \times 0.3048 = \mathbf{9.144\text{ m}}$
• Velocity ($U$): $12\text{ kn} \times 0.5144\text{ m/s per knot} = \mathbf{6.173\text{ m/s}}$

Calculate the characteristic Reynolds number based on the cylinder's diameter:

Step 2: calculating aerodynamic drag and towing power

Utilizing $C_{\text{D}} \approx 0.3$ from standard empirical data, calculate the total macroscopic drag force ($F_{\text{D}}$) acting on the projected rectangular area ($D \cdot L$):

Calculate the continuous mechanical power ($P$) required for the tow, which is the product of the drag force and the constant velocity:

Convert the result to horsepower ($1\text{ hp} \equiv 745.7\text{ W}$):

Step 3: vortex shedding frequency

Rearrange the Strouhal definition to solve for the absolute physical frequency in Hertz (cycles per second):

Principle in use: Dynamic Similarity. To accurately predict the performance of a full-scale prototype using a small-scale laboratory model, both systems must operate at identical dimensionless force ratios (e.g., Reynolds number). Only when dynamic similarity is achieved can force coefficients transfer directly between different fluids and scales.

Problem statement: A long square prism generates higher drag than a circular cylinder of comparable scale due to fixed flow separation at the sharp corners. Water-tunnel measurements on a square prism of side length $b = 2\text{ cm}$ ($0.02\text{ m}$) yield the following raw dimensional data:

(a) Use this laboratory data to predict the aerodynamic drag force per unit depth ($F/L$) of wind blowing at $6\text{ m/s}$, in standard air at $20^\circ\text{C}$, over a tall square chimney of side length $b = 55\text{ cm}$ ($0.55\text{ m}$).
(b) Is there any engineering uncertainty in your estimate? Justify your reasoning based on boundary layer mechanics.

Adapted from Fluid Mechanics by Frank M. White and Henry Xue.

Axiomatic setup: We utilize the Mass-Length-Time ($MLT$) continuum basis to establish primary dimensions.

Part (a): predicting drag via similarity

First, retrieve the standard fluid properties required for the calculations:
• For water (laboratory model): $\rho_w = 998\text{ kg/m}^3$, $\mu_w = 0.001\text{ kg/m}\cdot\text{s}$.
• For air (full-scale prototype): $\rho_a = 1.2\text{ kg/m}^3$, $\mu_a = 1.8 \times 10^{-5}\text{ kg/m}\cdot\text{s}$.

Convert the raw water tunnel data into dimensionless form. We compute $\mathrm{Re} = \frac{\rho_w V b}{\mu_w}$ and $C_F = \frac{F/L}{\rho_w V^2 b}$ for each discrete velocity (using $b=0.02\text{ m}$):

Across this entire evaluated Reynolds number range, the dimensionless force coefficient remains stable, averaging approximately $\mathbf{C_F \approx 1.055}$.

Assuming dynamic similarity holds for the prototype, we utilize this universal $C_F$ to predict the absolute drag force per unit depth ($F/L$) acting on the full-scale air chimney:

Part (b): evaluating extrapolation uncertainty

Calculate the Reynolds number for the full-scale chimney:

Yes, there is definitive uncertainty. The chimney operates at $\mathrm{Re} = 220{,}000$, which is $2.75\times$ higher than the maximum tested laboratory value of $\mathrm{Re} \approx 80{,}000$. Our prediction relies on the unverified assumption that $C_F$ remains constant at $1.055$ well beyond the experimental envelope. We are extrapolating, not interpolating.