Professor, what kind of problem is the backward-facing step flow?

It's a phenomenon where a flow separates and then reattaches at a step (a sudden change in geometry) within a flow channel. It has long been used as the most fundamental benchmark problem for separated and reattaching flows to validate CFD codes. The experimental data by Armaly et al. (1983) is particularly well-known.

So the image is that a vortex forms behind the step?

Correct. A recirculation zone (separation bubble) forms downstream of the step. The reattachment length, expressed as the ratio $x_r/h$ of the bubble length $x_r$ to the step height $h$, becomes a function of the Reynolds number. This is the most critical validation metric.

Backward-facing step flow

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

CAE visualization for backward facing step theory - technical simulation diagram

後向きステップ流れ

Theory and Physics

Overview

🧑‍🎓

Professor, what kind of problem is backward-facing step flow?

🎓

It's a phenomenon where there is a step (sudden change in height) in the middle of a flow channel, causing the flow to separate and then reattach. It is the most fundamental benchmark problem for separation and reattachment flows, and has long been used for verification of CFD codes. The experimental data by Armaly et al. (1983) is famous.

🧑‍🎓

So it's like a vortex forms behind the step?

🎓

Correct. A recirculation zone (separation bubble) forms downstream of the step. The reattachment length $x_r/h$, which is the length $x_r$ divided by the step height $h$, becomes a function of the Reynolds number. This is the most important verification metric.

Governing Equations

🧑‍🎓

The governing equations are the Navier-Stokes equations, right?

🎓

Yes, the incompressible Navier-Stokes equations and the continuity equation.

$$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla)\mathbf{u} = -\frac{1}{\rho}\nabla p + \nu \nabla^2 \mathbf{u} $$

$$ \nabla \cdot \mathbf{u} = 0 $$

🎓

The Reynolds number is defined using the step height $h$ and the inlet average velocity $U$.

$$ Re = \frac{Uh}{\nu} $$

🧑‍🎓

What is the relationship between the reattachment length and Re?

🎓

In the laminar flow regime (approximately $Re < 400$), the reattachment length is almost proportional to Re. An approximation $x_r/h \approx 0.06 \times Re$ is known. In Armaly et al.'s experiment with an expansion ratio $ER = (H+h)/H = 1.94$, $x_r/h \approx 5$ for Re=100 and $x_r/h \approx 14$ for Re=400.

🧑‍🎓

So the reattachment point moves further downstream as Re increases.

🎓

However, around $Re > 400$, three-dimensional effects become significant, and 2D calculations alone no longer match experiments. This is the famous "2D-3D transition problem of the Armaly problem."

Flow Structure

🧑‍🎓

Are there structures other than the recirculation zone?

🎓

As Re increases, a secondary separation bubble appears on the upper wall opposite the step. Furthermore, small vortices form at the step corner. The overall flow field structure strongly depends on Re.

Re Range	Flow Characteristics
Re < 200	Only the primary recirculation on the lower wall
200 < Re < 400	Secondary separation bubble appears on the upper wall
Re > 400	Three-dimensional instability, spanwise fluctuations
Re > 1000	Turbulent transition, unsteady vortex shedding

🧑‍🎓

So 2D calculations are sufficient only up to about Re=400. I see, thank you.

Coffee Break Yomoyama Talk

Why the Reattachment Point Position Becomes the "Litmus Test" for Turbulence Model Selection

In backward-facing step flow, the "distance from the step to where the flow reattaches to the wall downstream (reattachment length)" is a standard metric for comparing turbulence model performance. Experimental values show reattachment occurs at a position about 6–8 times the step height. However, solving with the standard k-ε model tends to delay reattachment to around 9–11 times, while using the SST model improves accuracy. This difference—"different answers for the same problem"—is the practical basis for selecting turbulence models. Despite its simple geometry, the backward-facing step contains all elements of "separation, recirculation, and reattachment," making it the first testbed when developing new turbulence models.

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 spluttering and unstable, 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—all are unsteady phenomena. So what is steady-state analysis? It looks only at "after sufficient time has passed and the flow has settled down"—meaning setting this term to zero. Since computational cost drops significantly, trying a steady-state solution first is a basic CFD strategy.
Convection Term $\nabla \cdot (\rho \mathbf{u} \phi)$: What happens if you drop a leaf into a river? 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 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 carried by flow, conduction is transmitted by molecules. There's 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 flows (slow, viscous), diffusion dominates. Conversely, in high Re flows, convection overwhelms, and diffusion plays a supporting role.
Pressure Term $-\nabla p$: When you push the plunger of a syringe, liquid shoots out forcefully from the needle, right? Why? Because the plunger side is high pressure, the needle tip is low pressure—this pressure difference provides the force that pushes the fluid. Dam discharge works on the same principle. On a weather map, where isobars are tightly packed? That's right, strong winds blow. "Flow is generated where there is a pressure difference"—this is the physical meaning of the pressure term in the Navier-Stokes equations. A point of confusion here: "Pressure" in CFD is often gauge pressure, not absolute pressure. If results become strange immediately after switching to compressible analysis, mixing up absolute/gauge pressure might be the cause.
Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it up. 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 an industrial 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 heated room 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, 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 gases (Kn > 0.1), supersonic/hypersonic flow (shock capturing required), free surface flow (VOF/Level Set, etc., required)

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: approx. 1.225 kg/m³ @20°C, Water: approx. 998 kg/m³ @20°C
Viscosity Coefficient $\mu$	Pa·s	Be careful not to confuse with kinematic viscosity $\nu = \mu/\rho$ [m²/s]
Reynolds Number $Re$	Dimensionless	$Re = \rho u L / \mu$. Criterion for laminar/turbulent transition
CFL Number	Dimensionless	$CFL = u \Delta t / \Delta x$. Directly related to time step stability

Numerical Methods and Implementation

Numerical Methods

🧑‍🎓

What should I be careful about in the numerical solution of backward-facing step flow?

🎓

For this problem, the pressure-velocity coupling is important. Use standard incompressible flow algorithms like SIMPLE-type or Projection methods.

Pressure-Velocity Coupling

🧑‍🎓

How does the SIMPLE method work?

🎓

It stands for Semi-Implicit Method for Pressure-Linked Equations, proposed by Patankar & Spalding (1972). The procedure is as follows.

1. Obtain a provisional velocity field from the momentum equations

2. Solve the pressure correction equation (Poisson-type)

3. Correct the velocity and pressure

4. Repeat until convergence

🎓

Derived methods include SIMPLEC, PISO (suited for unsteady). For steady calculations, SIMPLE/SIMPLEC is standard; for unsteady, PISO is standard.

Spatial Discretization

🧑‍🎓

What scheme is good for the convection term?

🎓

Since backward-facing step flow involves recirculation, using upwind differencing (1st order) causes excessive numerical diffusion, making the reattachment length too long. At least 2nd order accuracy is required.

Scheme	Accuracy	Stability	Effect on Reattachment Length
1st Order Upwind	1st	High	Overestimation (high numerical diffusion)
2nd Order Upwind	2nd	Medium	Appropriate
QUICK	3rd	Slightly Low	Appropriate
Central Differencing	2nd	Low	Risk of oscillations

Mesh Design

🧑‍🎓

Where should the mesh be refined?

🎓

Step Corner: Sufficient resolution is needed as it is the separation point

Wall Region Near Reattachment Zone: Refine in the wall-normal direction to capture where the wall shear stress changes sign

Outlet: Set sufficiently far downstream ($x_{out} \geq 30h$). If the outlet is too close, it affects the reattachment length

🎓

The $y^+$ of the first wall layer is not necessary for laminar flow, but for turbulent calculations, $y^+ < 1$ (without wall functions) is desirable. For structured grids, use an expansion ratio of 1.1–1.2 to distribute cells away from the wall.

🧑‍🎓

So placing the outlet far away is important. 30h, that's quite long.

🎓

If it's too short, the outlet boundary condition influences and changes the reattachment length. During verification, sensitivity to outlet position should also be checked.

Coffee Break Yomoyama Talk

The "Density Disparity" Problem in Mesh Immediately After the Step

A problem often overlooked in numerical calculations of backward-facing steps is the "mesh transition immediately after the step corner." The separation point (step edge) is where the velocity gradient is maximum, requiring a dense mesh. However, if the mesh becomes abruptly coarse from there towards the reattachment point, numerical diffusion increases, leading to over- or under-estimation of the recirculation zone. An empirical rule is "use element sizes less than 1/10 of the step height around the step edge and expand downstream with an expansion ratio of 1.1 or less" as a safe strategy. Also, even if the reattachment length matches experiments in a 2D model, that setting often doesn't hold for 3D models because 3D effects (corner flows) are ignored. It's safer not to carry over settings from "2D verification → 3D production."

Upwind Differencing (Upwind)

1st Order Upwind: High numerical diffusion but stable. 2nd Order Upwind: Improved accuracy but risk of oscillations. Essential for high Reynolds number flows.

Central Differencing (Central Differencing)

2nd order accuracy, but numerical oscillations occur for Pe number > 2. Suitable for low Reynolds number, diffusion-dominated flows.

TVD Schemes (MUSCL, QUICK, etc.)

Maintain high accuracy while suppressing numerical oscillations via limiter functions. Effective for capturing shocks and steep gradients.

Finite Volume Method vs Finite Element Method

FVM: Naturally satisfies conservation laws. Mainstream in CFD. FEM: Advantageous for complex shapes and multiphysics. Mesh-free methods like SPH are also developing.

CFL Condition (Courant Number)

Explicit methods: CFL ≤ 1 is the stability condition. Implicit methods: Stable even for CFL > 1, but affects accuracy and iteration count. LES: CFL ≈ 1 is recommended. Physical meaning: Information should not travel more than one cell per time step.

Residual Monitoring

Convergence is typically judged when residuals for continuity, momentum, and energy drop by 3–4 orders of magnitude. The mass conservation residual is particularly important.

Relaxation Factors

Pressure: 0.2–0.3, Velocity: 0.5–0.7 are typical initial values. If diverging, lower the relaxation factor. After convergence, increase to accelerate.

Internal Iterations for Unsteady Calculations

Iterate within each time step until a steady solution converges. Internal iteration count: 5–20 iterations is a guideline. If residuals fluctuate between time steps, review the time step size.

Analogy for the SIMPLE Method

The SIMPLE method is an "alternating adjustment" technique. First, obtain a provisional velocity (predictor step), then correct the pressure so that mass conservation is satisfied with that velocity (corrector step), then correct the velocity with the corrected pressure—repeat this back-and-forth to approach the correct solution. It resembles two people leveling a shelf: one adjusts the height, the other balances it, and they repeat this alternately.

Analogy for Upwind Differencing

Upwind differencing is a method that "stands in the river flow and prioritizes upstream information." A person in the river cannot tell where the water comes from by looking downstream—this discretization method reflects the physics that upstream information determines downstream conditions. Although it's 1st order accurate, it is highly stable because it correctly captures flow direction.