Professor, how does the flow around a sphere differ from that around a cylinder?

Because a sphere is a three-dimensional object, its wake structure is inherently 3D. There is no region where a 2D calculation suffices, unlike with a cylinder. Furthermore, the sharp change in the drag coefficient ($C_D$) with Reynolds number (Re), known as the drag crisis, is of great practical importance. The dimples on a golf ball utilize this phenomenon.

Could you start by explaining the most fundamental principles?

At Re $\ll 1$, the inertial terms can be neglected, and an exact solution to the Stokes equations exists. $$ \mathbf{F}_D = 6 \pi \mu a U_\infty $$ This is Stokes' drag law, where $a$ is the sphere's radius and $U_\infty$ is the freestream velocity. Rewriting it in terms of the drag coefficient yields: $$ C_D = \frac{F_D}{\frac{1}{2}\rho U_\infty^2 \pi a^2} = \frac{24}{Re}, \quad Re = \frac{2a U_\infty}{\nu} $$

Flow Around a Sphere

Q: So, $C_D = 24/Re$ means the drag coefficient decreases as Re increases.

However, the absolute drag force $F_D$ itself increases proportionally with $U_\infty$. The reason $C_D$ decreases is that the inertial force reference $\frac{1}{2}\rho U_\infty^2$ increases even more rapidly.

Category: 流体解析（CFD） | Integrated 2026-04-06

CAE visualization for flow around sphere theory - technical simulation diagram

球周りの流れ

Theory and Physics

Overview

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Professor, how is the flow around a sphere different from that around a cylinder?

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A sphere is a three-dimensional object, so its wake structure is inherently 3D. There is no region where a 2D calculation suffices, unlike with a cylinder. Also, the abrupt change in $C_D$ depending on Re, called the drag crisis, is practically very important. The dimples on a golf ball utilize this phenomenon.

Stokes Flow (Very Low Re)

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First, please explain the most basic part.

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When Re $\ll 1$, the inertial term can be neglected, and an exact solution to the Stokes equations exists.

$$ \mathbf{F}_D = 6 \pi \mu a U_\infty $$

This is Stokes' drag law. $a$ is the sphere's radius, $U_\infty$ is the freestream velocity. Rewriting it as a drag coefficient,

$$ C_D = \frac{F_D}{\frac{1}{2}\rho U_\infty^2 \pi a^2} = \frac{24}{Re}, \quad Re = \frac{2a U_\infty}{\nu} $$

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So $C_D = 24/Re$ means the drag coefficient decreases as Re increases, right?

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However, the absolute drag force $F_D$ increases proportionally with $U_\infty$. The reason $C_D$ decreases is that the inertial force reference $\frac{1}{2}\rho U_\infty^2$ increases even faster.

Flow Transition by Reynolds Number

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Let's organize the classification of flow around a sphere.

Re Range	Flow State	Characteristics
Re < 20	Steady Axisymmetric	No separation ~ tiny recirculation
20 < Re < 210	Steady Axisymmetric Wake	Growth of annular recirculation region
210 < Re < 270	Steady Non-axisymmetric	Loss of planar symmetry (regular bifurcation)
270 < Re < 800	Periodic Vortex Shedding	Regular shedding of hairpin vortices
800 < Re < $3 \times 10^5$	Subcritical Regime	Turbulent wake, $C_D \approx 0.44$
$3 \times 10^5$ < Re < $4 \times 10^5$	Drag Crisis	$C_D$ drops sharply from $0.44$ → $0.1$
Re > $4 \times 10^5$	Supercritical Regime	Turbulent boundary layer separation

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Compared to a cylinder, the Re at which vortex shedding begins is higher.

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Correct. Because a sphere is 3D, the flow can wrap around the object, delaying wake destabilization. Also, instead of a clear periodic pattern like a vortex street for a cylinder, the structure is more complex, involving the shedding of hairpin vortices.

Empirical Drag Correlation Formulas

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How do you estimate $C_D$ at intermediate Re numbers?

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The Schiller-Naumann correlation is commonly used.

$$ C_D = \frac{24}{Re}(1 + 0.15 Re^{0.687}), \quad Re < 800 $$

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For a wider range, there is Morrison's formula.

$$ C_D = \frac{24}{Re} + \frac{2.6(Re/5.0)}{1+(Re/5.0)^{1.52}} + \frac{0.411(Re/2.63\times10^5)^{-7.94}}{1+(Re/2.63\times10^5)^{-8.00}} + \frac{0.25(Re/10^6)}{1+(Re/10^6)} $$

This formula provides a continuous fit from $Re = 10^{-1}$ to $10^6$, including the drag crisis.

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Morrison's formula is quite complex. But it seems useful to be able to represent even the drag crisis.

Coffee Break Yomoyama Talk

The Paradox: Golf Ball Dimples Double the Flight Distance

If you launch a perfectly smooth sphere and a dimpled sphere at the same speed, which one flies farther? Intuition might suggest "the smooth one has less air resistance," but the answer is the opposite. The dimples induce boundary layer turbulence, pushing the separation point rearward, which drastically reduces the wake width. As a result, the drag coefficient becomes about half that of a smooth sphere. It is said that the flight distance of a typical golf ball is about twice that of a ball without dimples. This is a design that artificially induces the "drag crisis" at lower Re numbers, an aerodynamically elegant solution. When solving for a sphere's drag coefficient in CFD, the influence of surface roughness models is directly linked here.

Physical Meaning of Each Term

Temporal Term $\partial(\rho\phi)/\partial t$: Imagine the moment you turn on a faucet. At first, water comes out unstable and splashing, but after a while, it becomes a steady flow, right? This "period of change" is described by the temporal term. The pulsation of blood flow from a heartbeat, or the flow fluctuation each time an engine valve opens and closes, are all unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning this term is set to zero. This significantly reduces computational cost, so trying a steady-state solution first is a basic CFD strategy.
Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: If you drop a leaf into a river, what happens? It gets carried downstream by the flow, right? This is "convection"—the effect where fluid motion transports things. Warm air from a heater reaching the far corner of a room is also because the air, as a "carrier," transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow speed increases, this term rapidly strengthens, making control difficult. This is the root cause of turbulence. A common misconception: "Convection and conduction are similar" → They are completely different! Convection is transport by flow, conduction is transmission by molecules. There is an order of magnitude difference in efficiency.
Diffusion Term $\nabla \cdot (\Gamma \nabla \phi)$: Have you ever put milk in coffee and left it? Even without stirring, after a while it naturally mixes. That's molecular diffusion. Now a question—honey and water, which flows more easily? Obviously water, right? Honey has high viscosity ($\mu$), so it flows poorly. Higher viscosity strengthens the diffusion term, making the fluid move "sluggishly." In low Reynolds number flow (slow, viscous), diffusion dominates. Conversely, in high Re flow, convection overwhelms and diffusion plays a minor role.
Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle tip, right? Why? The piston side is high pressure, the needle tip is low pressure—this pressure difference provides the force pushing the fluid. Dam discharge works on the same principle. On a weather map, where isobars are densely packed? That's right, strong winds blow. "Flow arises where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A common point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results go wrong immediately after switching to compressible analysis, it might be due to mixing up absolute/gauge pressure.
Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, buoyancy pushes it upward. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat from a gas stove flame, Lorentz force acting on molten metal in a factory's electromagnetic pump... These are all actions that "inject energy or force into the fluid from the outside," expressed by the source term. What happens if you forget the source term? In natural convection analysis, forgetting buoyancy means the fluid doesn't move at all—a physically impossible result where warm air doesn't rise in a room with the heater on in winter.

Assumptions and Applicability Limits

Continuum Assumption: Valid for Knudsen number Kn < 0.01 (mean free path ≪ characteristic length)
Newtonian Fluid Assumption: Shear stress and strain rate have a linear relationship (non-Newtonian fluids require viscosity models)
Incompressibility Assumption (for Ma < 0.3): Density is treated as constant. For Mach number 0.3 and above, compressibility effects must be considered
Boussinesq Approximation (Natural Convection): Density variation is considered only in the buoyancy term; constant density is used in other terms
Non-applicable Cases: Rarefied gas (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (requires VOF/Level Set, etc.)

Dimensional Analysis and Unit Systems

Variable	SI Unit	Notes / Conversion Memo
Velocity $u$	m/s	When converting from volumetric flow rate for inlet conditions, pay attention to cross-sectional area units
Pressure $p$	Pa	Distinguish between gauge and absolute pressure. Use absolute pressure for compressible analysis
Density $\rho$	kg/m³	Air: ~1.225 kg/m³@20°C, Water: ~998 kg/m³@20°C
Viscosity Coefficient $\mu$	Pa·s	Be careful not to confuse with kinematic viscosity coefficient $\nu = \mu/\rho$ [m²/s]
Reynolds Number $Re$	Dimensionless	$Re = \rho u L / \mu$. Indicator for laminar/turbulent transition
CFL Number	Dimensionless	$CFL = u \Delta t / \Delta x$. Directly related to time step stability

Numerical Methods and Implementation

Selection of Numerical Methods

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What numerical method is suitable for calculating flow around a sphere?

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A sphere is inherently a 3D problem, so computational cost is significantly higher compared to a cylinder.

Re Range	Recommended Method	Mesh Scale Guideline
Re < 300	DNS	0.5 to 2 million cells
300 < Re < $10^4$	LES	5 to 50 million cells
$10^4$ < Re	RANS (SST) / DES	2 to 20 million cells

Mesh Strategy

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How do you create a mesh for a sphere?

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Using an O-type (spherical shell) mesh around the sphere surface is ideal.

First Wall Layer: $y^+ < 1$ (wall-resolved). For Re = 1000, $\Delta r / D \approx 5 \times 10^{-3}$
Spherical Direction Division: Finer near the equator (to track separation point movement), coarser near the poles is acceptable
Wake Region: Ensure at least $30D$ downstream from the sphere center. To track hairpin vortex development
Computational Domain Outer Radius: At least $20D$ from sphere center (blockage < 0.25%)

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How do you handle the singularity at the poles (the problem where mesh converges to a single point)?

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Good question. A standard spherical coordinate mesh collapses cells at the poles. Countermeasures include:

Cubed Sphere: Project each face of a cube onto a spherical surface. No polar singularity.
Unstructured Mesh: Prism layers near the sphere surface, tetrahedral/polyhedral elsewhere.
Overset Mesh: Cover the polar region with a separate patch.

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Practically, STAR-CCM+'s polyhedral mesh or Fluent's poly-hexcore are convenient. They automatically generate prism layers on the sphere surface and fill the outside with polyhedral cells.

Utilizing Axisymmetric Calculations

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For low Re, can we use an axisymmetric calculation?

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For the range of steady axisymmetric wake at Re < 210, calculation with axisymmetric (2D cross-section + no swirl) is possible. In OpenFOAM, you can run simpleFoam with a wedge mesh (5-degree wedge shape). Computational cost is less than 1/100th of 3D.

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However, for Re > 210, axisymmetry breaks, so full 3D calculation is absolutely necessary. It is dangerous to extrapolate to high Re just because "it worked with axisymmetric."

Coupling with Particle Tracking

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Flow around a sphere is related to particle sedimentation and such, right?

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Correct. The terminal velocity of a settling sphere from Stokes' law is,

$$ U_t = \frac{2(\rho_p - \rho_f) g a^2}{9 \mu} $$

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In CFD-DEM coupling, fluid forces (drag, lift, added mass force) are applied to particles. The Schiller-Naumann or Gidaspow models are standardly used as drag models. This can be implemented using Fluent's DPM module or OpenFOAM's DPMFoam.