Drag on bluff and streamlined bodies

Three of Jun's tasks this term turn on drag. The grit chamber at the head of the treatment plant must let sand settle while lighter organic matter stays in suspension, and its design depends on how fast grains fall through water. The new intake screen must withstand the river's push during spring floods. And the elevated water reservoir at Northside, a 16 m spherical tank on a 30 m column, is due for a structural review that includes a check of wind loads and of vortex-induced vibration of its column. All three need the same idea: the force a fluid exerts on a body moving relative to it, and how that force depends on the body's shape and the Reynolds number.

Drag and its two parts

The total force a fluid exerts on a body is the integral of pressure and shear stress over its surface. The component along the flow is drag; the component across it is lift, the subject of Lesson 7. Drag is written with a drag coefficient based on a reference area A, usually the frontal area for bluff bodies and the plan or wetted area for plates and wings:

Drag has two parts. Friction drag comes from wall shear and dominates for streamlined bodies and plates aligned with the flow, as in Lessons 4 and 5. Pressure drag, also called form drag, comes from the difference between high pressure on the front and low pressure in the separated wake behind. It dominates for bluff bodies, whose shapes force the boundary layer to separate. A flat plate across the flow has almost no friction drag, because the shear acts perpendicular to the flow direction, and almost all pressure drag. The same plate aligned with the flow has only friction drag, about a hundred times smaller.

Drag coefficients and Reynolds number

Dimensional analysis in Lesson 2 showed that the drag coefficient of a body of fixed shape depends on the Reynolds number and, for rough bodies, the relative roughness. At very low Reynolds number, the flow creeps around the body without separating and inertia is negligible. Stokes solved this case for a sphere in 1851:

As Re rises, the boundary layer separates and a wake forms; the drag coefficient falls more slowly and levels off at roughly 0.4 to 0.5 for a sphere between Re ≈ 10³ and 2 × 10⁵. Then something dramatic happens. Near Re ≈ 3 × 10⁵ for a smooth sphere, the boundary layer becomes turbulent before it separates, separation moves back from about 80° to about 120°, the wake narrows and the drag coefficient falls abruptly to about 0.1 to 0.2. This is the drag crisis. Circular cylinders show the same behavior, with C_D falling from about 1.2 to about 0.3 near Re ≈ 3 × 10⁵ and recovering partly at higher Re. Roughness moves the crisis to a lower Reynolds number, which is what the dimples on a golf ball accomplish.

Sharp-edged bodies, such as plates and square sections, separate at their edges whatever the Reynolds number, so their drag coefficients hardly change once Re is above about 10³. Rounded bodies are sensitive, because the separation point moves. This distinction is why building codes give fixed force coefficients for most structural shapes but treat round chimneys, tanks and towers with care.

Settling grit: drag at low Reynolds number

A grain falling steadily through still water reaches its terminal velocity when its weight, less buoyancy, equals its drag:

Because C_D depends on the Reynolds number, which depends on V_t, the equation is solved by iteration. The program starts from Stokes' law and repeats using a standard curve fit for the sphere drag coefficient until the speed converges. Grit chambers are commonly designed to remove sand grains of about 0.2 mm and larger with a specific gravity near 2.65.

# Drag on bluff bodies: grit settling, an intake screen, and wind on the elevated water tower
import math
g = 9.81
def cd_sphere(Re):                 # standard drag curve fit, valid to about Re = 2e5
    return 24 / Re * (1 + 0.15 * Re**0.687) + 0.42 / (1 + 4.25e4 * Re**-1.16)
# 1. Terminal settling speed of sand grains in the grit chamber (water at 10 C)
rho_w, mu_w, rho_s = 999.7, 1.31e-3, 2650.0
for d in (0.10e-3, 0.21e-3, 0.50e-3, 1.0e-3):
    v = (rho_s - rho_w) * g * d**2 / (18 * mu_w)          # start from Stokes' law
    for _ in range(100):
        Re = rho_w * v * d / mu_w
        v = math.sqrt(4 * (rho_s - rho_w) * g * d / (3 * cd_sphere(Re) * rho_w))
    Re = rho_w * v * d / mu_w
    print(f"d = {d*1000:.2f} mm: settling speed {v*1000:5.1f} mm/s, Re = {Re:5.1f}, Stokes would give {(rho_s-rho_w)*g*d**2/(18*mu_w)*1000:6.1f} mm/s")

# 2. Cylindrical wedge-wire intake screen in the river current
D, Ls, V = 0.9, 2.4, 0.30
Re = rho_w * V * D / mu_w
Cd = 1.2                           # subcritical circular cylinder
print(f"\nScreen: Re = {Re:.1e}, drag {Cd*0.5*rho_w*V**2*D*Ls:.0f} N; during ice-free flood at 1.5 m/s {Cd*0.5*rho_w*1.5**2*D*Ls:.0f} N")

# 3. Wind on the elevated water tower: air density from local station pressure
p, T = 97.4e3, 273.15 - 10.0     # station pressure and a cold, windy day
rho_a = p / (287.0 * T)
mu_a = 1.68e-5
V = 35.0                           # design gust speed used for this check, m/s
D_tank, D_col, H_col = 16.0, 3.0, 30.0
Re_t = rho_a * V * D_tank / mu_a
F_tank = 0.2 * 0.5 * rho_a * V**2 * math.pi * D_tank**2 / 4   # supercritical sphere, C_D about 0.2
F_tank_sub = 0.47 * 0.5 * rho_a * V**2 * math.pi * D_tank**2 / 4
F_col = 0.7 * 0.5 * rho_a * V**2 * D_col * H_col                # supercritical cylinder, C_D about 0.7
print(f"\nAir density {rho_a:.3f} kg/m3; tank Reynolds number {Re_t:.1e}")
print(f"Tank drag {F_tank/1000:.0f} kN (C_D 0.2); it would be {F_tank_sub/1000:.0f} kN at the subcritical C_D 0.47")
print(f"Column drag {F_col/1000:.0f} kN; overturning moment at the base {(F_tank*(H_col+D_tank/2) + F_col*H_col/2)/1000:,.0f} kN.m")
# 4. Vortex shedding from the column: wind speed that matches its natural frequency
St, fn = 0.20, 1.1                 # Strouhal number and the column's first natural frequency, Hz
V_cr = fn * D_col / St
print(f"Shedding frequency at 35 m/s {St*V/D_col:.2f} Hz; lock-in wind speed for f_n = {fn} Hz is {V_cr:.1f} m/s")

d = 0.10 mm: settling speed   6.3 mm/s, Re =   0.5, Stokes would give    6.9 mm/s
d = 0.21 mm: settling speed  22.3 mm/s, Re =   3.6, Stokes would give   30.3 mm/s
d = 0.50 mm: settling speed  70.4 mm/s, Re =  26.9, Stokes would give  171.6 mm/s
d = 1.00 mm: settling speed 143.4 mm/s, Re = 109.5, Stokes would give  686.6 mm/s

Screen: Re = 2.1e+05, drag 117 N; during ice-free flood at 1.5 m/s 2915 N

Air density 1.290 kg/m3; tank Reynolds number 4.3e+07
Tank drag 32 kN (C_D 0.2); it would be 75 kN at the subcritical C_D 0.47
Column drag 50 kN; overturning moment at the base 1,954 kN.m
Shedding frequency at 35 m/s 2.33 Hz; lock-in wind speed for f_n = 1.1 Hz is 16.5 m/s

The 0.21 mm design grain settles at about 22 mm/s in 10 °C water. Stokes' law alone would overestimate that by more than a third, because the grain's Reynolds number of 3.6 is already beyond the creeping-flow range. For a 1 mm grain the Stokes estimate is almost five times too high. The chamber's length is set by the time a grain needs to fall through the water depth while the horizontal flow carries it along, so an error in settling speed translates directly into a chamber that is too short. Cold water makes the problem worse: viscosity at 2 °C is about 30 percent higher than at 10 °C, and grains settle more slowly in winter.

The intake screen in the river

The intake screen of Lesson 3 is a cylinder 0.9 m in diameter and 2.4 m long. In the normal 0.30 m/s current its Reynolds number is about 2 × 10⁵, just below the drag crisis, so a subcritical drag coefficient of about 1.2 applies and the drag is only about 120 N. In a spring flood at 1.5 m/s the drag rises with the square of the speed to about 2.9 kN, before allowing for debris. Hana asks Jun to design the screen's support for a much larger load, because debris mats and ice can block the screen, turning it into something closer to a solid flat plate with a larger frontal area and a drag coefficient near 2. The fluid mechanics sets the clean-screen load; engineering judgment sets the design load.

Wind on the elevated reservoir

The Northside elevated reservoir has a spherical tank 16 m in diameter on a cylindrical column 3 m in diameter and 30 m high. The structural engineer will use the building code's wind provisions for the final check, but Hana wants an independent estimate from first principles. On a cold, windy day the air is denser than the standard value, and the program finds the density from Omaha's station pressure. At a 35 m/s gust the tank's Reynolds number is about 4 × 10⁷, far above the drag crisis, so the supercritical coefficient of about 0.2 applies to a smooth tank.

The program estimates a tank drag of 32 kN with C_D = 0.2, but 75 kN if the flow behaved as subcritical. The difference shows why the drag crisis matters in design: a model test at too low a Reynolds number would overstate the load by a factor of more than two unless the model were roughened to trip its boundary layer. The column's drag adds about 50 kN, and together they produce an overturning moment at the base of about 1,950 kN·m. The structural engineer will compare this with the code's value, which includes gust effects, the variation of wind speed with height and importance factors.

In the United States, wind loads are calculated from ASCE 7, which gives basic wind speeds on maps by risk category and force coefficients for common shapes, and which most state building codes adopt through the International Building Code. Elevated water tanks are also commonly designed to AWWA D100 for welded steel tanks, which has its own wind provisions.

Vortex shedding

Behind a bluff body, separation does not produce a steady wake. Over a wide range of Reynolds numbers, vortices are shed alternately from each side, forming a regular pattern called a Kármán vortex street. Each vortex shed produces a side force on the body, so the body feels an alternating cross-flow force at the shedding frequency f, which is given by the Strouhal number:

If the shedding frequency approaches a natural frequency of the structure, the vortices and the structure's motion can lock together, and the vibration grows much larger than the force alone would suggest. This is vortex-induced vibration. It sings in power lines, it shook the first tall steel chimneys, and it is a design check for slender columns, risers and pipelines in currents. The reservoir column's first natural frequency is about 1.1 Hz, and the program finds that shedding at that frequency occurs at a wind speed of about 16.5 m/s, a common wind. The structure's damping and mass make the response small for this heavy column, but the check is required, and slender steel stacks are often fitted with helical strakes that break up the coherent shedding.

Streamlining

Streamlining reduces drag by shaping the body so that its boundary layer stays attached as the pressure recovers toward the rear. A well-streamlined strut, with a rounded nose and a long tapering tail, can have a drag coefficient based on frontal area of about 0.06, compared with 1.2 for a circular cylinder of the same thickness: twenty times less. Its friction drag is higher, because it has more surface, but its pressure drag nearly vanishes. The trade-off is length, cost and sensitivity to flow direction: a streamlined strut works only when the flow approaches along its axis. River piers are often given rounded or pointed noses for this reason, and the bars of the intake's trash rack are shaped like flattened teardrops to lower their head loss.

Drag of vehicles and people

Drag coefficients for vehicles range from about 0.25 for a modern car to 0.6 or more for a pickup truck or a van, and 0.6 to 0.9 for heavy trucks without fairings. The crew's service truck, with a frontal area of about 3 m² and C_D ≈ 0.55, needs about 21 kW at the wheels just to overcome air drag at 100 km/h, roughly half its total road load. For workers on the reservoir's catwalk, a standing person has a drag area C_D A of about 0.8 m², so a 25 m/s gust pushes with about 300 N, enough to make a fall-protection harness more than a formality.