cases:case062

# Developing Flow in a Curved Rectangular Duct

The flow considered is a developing turbulent flow in a 90 deg. curved duct of rectangular cross-section. As shown in figure 1 there is an initial 2D wind tunnel constriction, resulting in a rectangular duct cross-section of width $H=20.3$ cm and height $6H$. Upstream of the bend there is a straight section of duct, of length $7.5H$, and another straight section downstream of the bend, of length $25.5H$. The bend itself has inner and outer radii of $R_i=3H$ and $R_o=4H$.

Fig. 1: Schematic of duct geometry

Measurements have been taken at two upstream locations (stations U1 and U2), three positions around the bend (15, 45 and 75) and two locations downstream of the bend (D1 and D2). Details are given in the table below.

Measurement station Location
U1 $4.5H$ upstream of the bend
U1 $0.5H$ upstream of the bend
15 $15^o$ into the bend
45 $45^o$ into the bend
75 $75^o$ into the bend
D1 $0.5H$ downstream of the bend
D2 $4.5H$ downstream of the bend

#### Flow Characteristics

The initially 2D boundary layers developing on the vertical lateral walls are subjected to strong streamwise curvatures and associated pressure gradients along the bend. On the other hand, the pressure-driven secondary motion in the corner regions eventually leads to the formation of a longitudinal vortex on the convex wall. The duct aspect ratio is such that these two features of the flow develop more or less independently, without interaction.

#### Flow Parameters

• Air with a kinematic viscosity: $\nu = 1.45 \times 10^{-5}$ m2/s.
• Freestream velocity at station $U1$ ($x = -4.5H$): $U_o = 16$ m/s.
• Reynolds number: $U_oH/\nu = 224,000$.

#### Inflow Conditions

At station U1 ($x = -4.5H$), the velocity is uniform in the core flow, outside the boundary layers, within a deviation less than 1%. On the vertical lateral walls, the boundary layers are of flat-plate type with a momentum thickness Reynolds number of 1650, a boundary layer thickness of $\delta = 0.08H$ and a friction coefficient of $C_f = 0.0038$. The 2D wind-tunnel contraction located $3H$ upstream of U1 does introduce a secondary motion in the boundary layers on the top and bottom flat walls, but its magnitude reaches only 5% of the freestream velocity. Measurements are provided for what is the slightly three-dimensional duct flow at this location, which should in most cases be sufficiently detailed to be used as inlet conditions.

Hot-wire velocity measurements have been carried out using a miniature X-wire probe for the turbulence quantities.

Mean velocity measurements have been carried out using a five-hole pressure probe with a diameter of 3 mm.

All velocity measurements have been made in the upper half of the duct divided into 5 different domains, namely in1, up1, ou1, in2, ou2 (see figure 2).

Wall shear stress, $\tau_w$, measurements were made using two pressure probes in combination (only the magnitude is actually measured). The friction coefficient is defined as $C_f = 2\tau_w/(\rho U_o^2)$.

Static pressure measurements were made using wall taps. The pressure coefficient is defined as $C_p = 2(p - p_o)/(\rho U_o^2)$, where $p_o$ is the static pressure at ($0$,$0$,$3H$).

Fig. 2: Measurement domains across the upper half of the duct cross section

#### Measurement Errors

 $\delta(U)$ 1.5% $\delta(V)$, $\delta(W)$ 3% $\delta(\overline{u^2})$ 5% $\delta(\text{other Reynolds stresses})$ 10%

The measurements available include:

• Contour maps at selected streamwise locations (fice files per location, covering the five regions indicated in figure ##):
• Mean velocities $U$, $V$ and $W$ at stations U1, U2, 15, 45, 75, D1, and D2
• Reynolds stresses $\overline{u^2}$, $\overline{v^2}$, $\overline{w^2}$, $\overline{uv}$, $\overline{uw}$ at stations U1, U2, 45 and D1
• Profiles across the duct at a number of $Z$ locations:
• Mean velocities $U$, $V$ and $W$ at stations U1, U2, 15, 45, 75, D1, and D2
• Reynolds stresses $\overline{u^2}$, $\overline{v^2}$, $\overline{w^2}$, $\overline{uv}$, $\overline{uw}$ at stations U1, U2, 45 and D1
• Wall pressure coefficient around the inner and outer walls of the duct, at duct mid-height
• Skin friction coefficient around the duct walls at measurement stations U1, U2, 15, 45, 75, D1 and D2

Sample plots of selected quantities are available.

The file readme.txt has information on the data file naming and contents.

Pressure coefficients:

 pressure.dat $C_p$ around inner and outer duct walls pressure-tab.dat The same $C_p$ data, but tabulated for different parts of inner/outer walls

Skin friction:

Contour maps:

Cross-Duct Profiles:

Station Mean Velocities Reynolds Stresses
U1 mu1sel.dat tu1sel.dat
U2 mu2sel.dat tu2sel.dat
15 m15sel.dat
45 m45sel.dat t45sel.dat
75 m75sel.dat
D1 md1sel.dat td1sel.dat
D2 md2sel.dat

#### Recommendations for calculations

Inlet conditions: The calculation of the duct flow can be started at station $U1$ using the experimental values provided as inlet conditions. The non-measured quantity $\overline{vw}$ may be assumed as negligible.

Symmetry: Due to geometric symmetry with respect to the $z=0$ plane, one can use a computational domain including only the upper half of the duct.

Outlet conditions: The outlet should be placed sufficiently far downstream of the bend ($x > 30H$) so that zero gradients may be assumed for the flow variables.

Sotiropoulos and Patel (1992) have performed calculations of this case with the two-layer $k$-$\varepsilon$ model using two different numerical methods: the “finite-analytic” and a finite-difference method. In both cases, the overall structure of the flow was well predicted but both the strength of the secondary motion, and consequently its effect on the streamwise flow development, and the effects of wall curvature on the turbulence within the lateral boundary layers were underestimated.

1. Kim, W.J., Patel, V.C. (1994). Origin and decay of longitudinal vortices in developing flow in a curved rectangular duct. J. Fluids Engineering, Vol. 116, pp. 45-52.
2. Sotiropoulos, F., Patel, V.C. (1992). Flow in curved ducts of varying cross-section. IIHR Report No. 358, Institute of Hydraulic Research, University of Iowa.

Indexed data:

case062 (dbcase, confined_flow)
case062
titleDeveloping flow in curved rectangular duct
authorKim, Patel
year1991
typeEXP
flow_tag3d, curvature, constant_cross_section