Chapter 16.1: Canyon Flow Regimes

16.1.1 Street Canyons as Fluid Domains

A street canyon is the quasi-two-dimensional void bounded by building facades on both sides of a road and the ground surface. When ambient wind flows over the rooftop level, it interacts with these walls to produce characteristic recirculation patterns that govern pollutant dispersion at pedestrian breathing height.

The single most important geometric parameter is the aspect ratio:

$$\alpha = \frac{H}{W}$$

where $H$ is the mean building height and $W$ is the canyon width (facade-to-facade distance). Wind-tunnel experiments by Oke (1988) and subsequent CFD studies established that $\alpha$ alone classifies the flow into four distinct regimes.

Understanding these regimes is essential for predicting where pollutants accumulate, how long they persist, and which pedestrians are most exposed. This chapter derives the governing physics from first principles.

16.1.2 The Four Flow Regimes

Regime 1: Isolated Roughness (alpha < 0.3)

When buildings are far apart relative to their height, the wake shed by the upwind building dissipates before reaching the downwind building. Each building behaves as an isolated bluff body. The separation bubble reattaches to the ground surface well within the canyon, leaving the downwind building unaffected. No coherent vortex spans the full canyon width.

The flow reattachment length scales as $x_r \approx 4\text{--}6\,H$ for sharp-edged buildings. Since $W > H/0.3 \approx 3.3\,H$, the wake has room to recover. Pollutants emitted at street level are efficiently ventilated upward through the open top.

Regime 2: Wake Interference (0.3 < alpha < 0.7)

As the canyon narrows, the wake of the upwind building begins to impinge on the downwind facade before fully reattaching. The two wakes interfere, creating unsteady, asymmetric flow patterns. A weak, intermittent vortex may form but is not stable. This transitional regime exhibits the highest temporal variability in ground-level concentrations.

Regime 3: Skimming Flow (0.7 < alpha < 1.5)

For typical urban canyons, the ambient flow “skims” over the rooftop level, driving a single, stable, quasi-two-dimensional vortex inside the canyon. This is the most common and most studied regime. The vortex rotates such that air descends along the downwind facade and rises along the upwind facade. Street-level pollutants on the leeward side are trapped beneath the vortex core, producing the characteristic asymmetry in concentration fields: the leeward wall is dirtier than the windward wall.

$$\text{Skimming flow criterion:} \quad \frac{H}{W} \gtrsim 0.7 \quad \Longrightarrow \quad \text{stable single vortex}$$

Regime 4: Deep Canyon (alpha > 1.5)

When the canyon is very narrow relative to building height, the single vortex can no longer fill the entire cross-section. A second counter-rotating vortex forms below the primary one. For very deep canyons ($\alpha > 3$), a third vortex may appear. The lower vortices are progressively weaker, meaning pollutants emitted at ground level have very long residence times—these are the most hazardous configurations.

$$n_{\text{vortices}} \approx \left\lfloor \alpha \right\rfloor \quad \text{for } \alpha > 1.5$$

16.1.3 Regime Characteristics Summary

Regime	alpha Range	Vortex Structure	Ventilation	Pollution Risk
Isolated Roughness	< 0.3	None (wake reattaches)	Excellent	Low
Wake Interference	0.3 – 0.7	Intermittent / weak	Good but variable	Moderate
Skimming Flow	0.7 – 1.5	Single stable vortex	Poor (leeward side)	High
Deep Canyon	> 1.5	Multiple counter-rotating	Very poor at ground	Very High

16.1.4 Vortex Circulation and Velocity

In the skimming-flow regime, the ambient wind at rooftop height $u_H$ drives the canyon vortex through shear stress transfer at the canyon-atmosphere interface. The characteristic velocity inside the vortex can be derived from a momentum balance.

Momentum Transfer at Roof Level

The shear stress at the rooftop interface is:

$$\tau_{\text{roof}} = \rho \, C_D \, u_H^2$$

where $C_D$ is a drag coefficient characterising the roughness of the roof-level shear layer (typically 0.001–0.01 depending on building geometry).

This stress acts over the canyon width $W$ and is balanced by frictional drag on the canyon walls and floor. The wall drag per unit length scales as $\rho \kappa u_v^2$, where $\kappa \approx 0.41$ is the von Karman constant and $u_v$ is the characteristic vortex velocity. The total wall perimeter is approximately $W + 2H = W(1 + 2\alpha)$.

Equilibrium Balance

Balancing momentum input (roof shear) against wall friction dissipation:

$$\rho \, C_D \, u_H^2 \cdot W = \rho \, \kappa \, u_v^2 \cdot (W + 2H)$$

Solving for the vortex velocity:

$$u_v = \sqrt{\frac{C_D}{\kappa}} \cdot \frac{u_H}{\sqrt{1 + 2\alpha}}$$

For a simplified scaling (using a single factor in the denominator), this is often written as:

$$u_v = \frac{C_D \, u_H}{(1 + \alpha)\,\kappa}$$

This simplified form treats the drag-shear transfer as linear and absorbs the square-root into the effective drag coefficient. Both forms capture the essential physics: the vortex velocitydecreases with increasing aspect ratio because deeper canyons have more wall surface to absorb momentum.

Vortex Circulation

The circulation of the canyon vortex is defined as the line integral of velocity around the vortex perimeter:

$$\Gamma = \oint \mathbf{u} \cdot d\mathbf{l} \approx u_v \cdot (2H + 2W) = 2\,u_v\,W\,(1 + \alpha)$$

Substituting the expression for $u_v$:

$$\Gamma = \frac{2\,C_D\,u_H\,W}{\kappa}$$

Remarkably, the circulation is independent of aspect ratio in this model—deeper canyons have slower vortex velocities but larger perimeters, and the two effects cancel exactly.

16.1.5 Above-Roof Velocity Profile

Above the canyon, the mean wind profile follows the standard logarithmic law of the atmospheric surface layer:

$$u(z) = \frac{u_*}{\kappa} \ln\!\left(\frac{z - d}{z_0}\right)$$

where $u_*$ is the friction velocity, $d \approx 0.7H$ is the displacement height, and $z_0 \approx 0.1H$ is the aerodynamic roughness length. The rooftop velocity $u_H = u(H)$ then becomes:

$$u_H = \frac{u_*}{\kappa} \ln\!\left(\frac{0.3H}{0.1H}\right) = \frac{u_*}{\kappa}\ln 3 \approx 2.68\,u_*$$

This connects the mesoscale wind field to the rooftop forcing that drives the canyon vortex.

16.1.6 Python: Visualise Flow Regimes and Vortex Velocity

The following code visualises the four flow regimes as schematic cross-sections and plots the vortex velocity $u_v$ as a function of aspect ratio $\alpha$.

Canyon Flow Regimes and Vortex Velocity

Python

script.py162 lines

import numpy as np
import matplotlib.pyplot as plt
import matplotlib.patches as patches
from matplotlib.patches import FancyArrowPatch

# ─── Parameters ───
kappa = 0.41       # von Karman constant
C_D = 0.005        # drag coefficient
u_H = 5.0          # rooftop wind speed (m/s)

# ─── Figure 1: Schematic of Four Flow Regimes ───
fig, axes = plt.subplots(1, 4, figsize=(16, 4))
regimes = [
    {"name": "Isolated\nRoughness", "alpha": 0.2, "color": "#22c55e"},
    {"name": "Wake\nInterference", "alpha": 0.5, "color": "#eab308"},
    {"name": "Skimming\nFlow", "alpha": 1.0, "color": "#f97316"},
    {"name": "Deep\nCanyon", "alpha": 2.0, "color": "#ef4444"},
]

for ax, reg in zip(axes, regimes):
    alpha = reg["alpha"]
    W = 2.0  # normalised canyon width
    H = alpha * W

# Draw buildings
    ax.add_patch(patches.Rectangle((-W/2 - 0.6, 0), 0.6, H,
                 facecolor="#334155", edgecolor="#64748b", linewidth=1.5))
    ax.add_patch(patches.Rectangle((W/2, 0), 0.6, H,
                 facecolor="#334155", edgecolor="#64748b", linewidth=1.5))
    # Ground
    ax.plot([-W/2 - 0.6, W/2 + 0.6], [0, 0], color="#64748b", linewidth=2)

# Roof-level wind arrow
    ax.annotate("", xy=(W/2 + 0.6, H + 0.15), xytext=(-W/2 - 0.6, H + 0.15),
                arrowprops=dict(arrowstyle="->", color="#10b981", lw=2))
    ax.text(0, H + 0.3, f"u_H", ha="center", va="bottom", color="#10b981", fontsize=9)

# Vortex indicators
    if alpha < 0.3:
        # Isolated: small wake behind left building
        theta = np.linspace(0, np.pi, 30)
        r = 0.3 * H
        cx, cy = -W/2 + r + 0.1, r
        ax.plot(cx + r*np.cos(theta), cy + r*np.sin(theta), color=reg["color"], lw=1.5, ls="--")
    elif alpha < 0.7:
        # Wake interference: partial, unstable vortex
        theta = np.linspace(0.3, 2*np.pi - 0.3, 40)
        rx, ry = W/2 * 0.6, H * 0.5
        ax.plot(rx*np.cos(theta), ry*np.sin(theta) + H*0.4,
                color=reg["color"], lw=1.5, ls="--")
    elif alpha < 1.5:
        # Skimming: single full vortex
        theta = np.linspace(0, 2*np.pi, 60)
        rx, ry = W/2 * 0.7, H * 0.4
        ax.plot(rx*np.cos(theta), ry*np.sin(theta) + H*0.5,
                color=reg["color"], lw=2)
        # rotation arrow
        ax.annotate("", xy=(0.1, H*0.85), xytext=(0.4, H*0.9),
                    arrowprops=dict(arrowstyle="->", color=reg["color"], lw=1.5))
    else:
        # Deep canyon: two vortices
        for i, frac in enumerate([0.7, 0.3]):
            theta = np.linspace(0, 2*np.pi, 60)
            rx, ry = W/2 * 0.6, H * 0.2
            cy = H * frac
            ax.plot(rx*np.cos(theta), ry*np.sin(theta) + cy,
                    color=reg["color"], lw=2, ls=("-" if i == 0 else "--"))

ax.set_xlim(-W/2 - 1, W/2 + 1)
    ax.set_ylim(-0.3, max(H + 0.8, 1.5))
    ax.set_aspect("equal")
    ax.set_title(reg["name"], fontsize=10, color=reg["color"], fontweight="bold")
    ax.text(0, -0.2, f"α = {alpha}", ha="center", fontsize=9, color="white")
    ax.set_facecolor("#0f172a")
    ax.tick_params(colors="gray", labelsize=7)
    for spine in ax.spines.values():
        spine.set_color("#334155")

fig.patch.set_facecolor("#0f172a")
plt.suptitle("Street Canyon Flow Regimes", color="white", fontsize=14, fontweight="bold", y=1.02)
plt.tight_layout()
plt.savefig("output.png", dpi=120, bbox_inches="tight", facecolor="#0f172a")
plt.close()

# ─── Figure 2: Vortex Velocity vs Aspect Ratio ───
fig2, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

alpha_arr = np.linspace(0.05, 4.0, 200)

# Simplified model: u_v = C_D * u_H / ((1 + alpha) * kappa)
u_v_simple = C_D * u_H / ((1 + alpha_arr) * kappa)

# Full model: u_v = sqrt(C_D / kappa) * u_H / sqrt(1 + 2*alpha)
u_v_full = np.sqrt(C_D / kappa) * u_H / np.sqrt(1 + 2 * alpha_arr)

# Circulation (independent of alpha in simplified model)
Gamma = 2 * C_D * u_H * 10.0 / kappa  # W = 10m example
Gamma_arr = np.ones_like(alpha_arr) * Gamma

# Plot vortex velocity
ax1.plot(alpha_arr, u_v_simple, color="#10b981", linewidth=2.5, label="Simplified: $C_D u_H / [(1+\\alpha)\\kappa]$")
ax1.plot(alpha_arr, u_v_full, color="#06b6d4", linewidth=2.5, linestyle="--",
         label="Full: $\\sqrt{C_D/\\kappa}\\, u_H / \\sqrt{1+2\\alpha}$")

# Regime boundaries
for boundary, label, c in [(0.3, "Isolated→Wake", "#22c55e"),
                             (0.7, "Wake→Skimming", "#eab308"),
                             (1.5, "Skimming→Deep", "#f97316")]:
    ax1.axvline(boundary, color=c, linestyle=":", alpha=0.7, linewidth=1.5)
    ax1.text(boundary + 0.05, ax1.get_ylim()[0] if ax1.get_ylim()[0] > 0 else 0.001,
             label, rotation=90, va="bottom", fontsize=8, color=c)

ax1.set_xlabel("Aspect Ratio α = H/W", color="white", fontsize=12)
ax1.set_ylabel("Vortex Velocity u_v (m/s)", color="white", fontsize=12)
ax1.set_title("Vortex Velocity vs Aspect Ratio", color="white", fontsize=13, fontweight="bold")
ax1.legend(fontsize=9, facecolor="#1e293b", edgecolor="#334155", labelcolor="white")
ax1.set_facecolor("#0f172a")
ax1.tick_params(colors="gray")
ax1.grid(True, alpha=0.2, color="#334155")
for spine in ax1.spines.values():
    spine.set_color("#334155")

# Plot ventilation time scale
W_val = 10.0  # 10m canyon width
tau_vent = W_val / (u_v_simple * (1 + alpha_arr))  # ventilation timescale
# Cap for visualization
tau_vent = np.clip(tau_vent, 0, 1000)

ax2.plot(alpha_arr, tau_vent, color="#f59e0b", linewidth=2.5)
ax2.set_xlabel("Aspect Ratio α = H/W", color="white", fontsize=12)
ax2.set_ylabel("Ventilation Time Scale (s)", color="white", fontsize=12)
ax2.set_title("Canyon Ventilation Time (W = 10 m)", color="white", fontsize=13, fontweight="bold")
ax2.set_facecolor("#0f172a")
ax2.tick_params(colors="gray")
ax2.grid(True, alpha=0.2, color="#334155")
ax2.set_ylim(0, 500)
for spine in ax2.spines.values():
    spine.set_color("#334155")

# Regime shading
for ax in [ax1, ax2]:
    ax.axvspan(0, 0.3, alpha=0.1, color="#22c55e")
    ax.axvspan(0.3, 0.7, alpha=0.1, color="#eab308")
    ax.axvspan(0.7, 1.5, alpha=0.1, color="#f97316")
    ax.axvspan(1.5, 4.0, alpha=0.1, color="#ef4444")

fig2.patch.set_facecolor("#0f172a")
plt.tight_layout()
plt.savefig("output.png", dpi=120, bbox_inches="tight", facecolor="#0f172a")
plt.close()

print("=== Canyon Flow Regime Analysis ===")
print(f"Parameters: C_D = {C_D}, κ = {kappa}, u_H = {u_H} m/s")
print()
for a in [0.2, 0.5, 1.0, 1.5, 2.0, 3.0]:
    uv_s = C_D * u_H / ((1 + a) * kappa)
    uv_f = np.sqrt(C_D / kappa) * u_H / np.sqrt(1 + 2*a)
    regime = ("Isolated" if a < 0.3 else
              "Wake Interference" if a < 0.7 else
              "Skimming" if a < 1.5 else "Deep Canyon")
    print(f"α = {a:.1f} ({regime:>18s}): u_v(simple) = {uv_s:.4f} m/s, u_v(full) = {uv_f:.4f} m/s")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

16.1.7 Key Takeaways

The aspect ratio $\alpha = H/W$ is the primary control parameter for street canyon aerodynamics.
Four regimes exist: isolated roughness, wake interference, skimming flow, and deep canyon. Each has qualitatively different pollutant dispersion characteristics.
Vortex velocity decreases with aspect ratio: $u_v \propto 1/(1 + \alpha)$. Deeper canyons have slower ventilation.
The circulation $\Gamma$ is approximately independent of aspect ratio in the simplified momentum balance.
The von Karman constant $\kappa \approx 0.41$ appears throughout because wall turbulence governs the momentum transfer.

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