Jet Flow
Theory and Physics
Overview
Teacher, a jet is essentially the flow coming out of a nozzle, right?
That's correct. A jet is a flow discharged from a nozzle or orifice into a surrounding fluid. Its industrial applications are wide-ranging. From jet engine exhaust, welding torches, air conditioning vents, chemical plant mixers, to inkjet printers.
From a fluid dynamics perspective, a jet is a representative example of free shear flow, and along with mixing layers and wakes, it is a fundamental subject for turbulence research.
Jet Classification
Are there different types of jets?
Classified by geometric shape, they are as follows.
| Type | Shape | Velocity Decay in Self-Similar Region | Spread Rate |
|---|---|---|---|
| Axisymmetric Circular Jet | Circular Nozzle | $u_c / U_0 \propto (x/D)^{-1}$ | $\delta / x \approx 0.10$ |
| Plane Jet | Slit Nozzle | $u_c / U_0 \propto (x/h)^{-1/2}$ | $\delta / x \approx 0.11$ |
| Rectangular Jet | Rectangular Nozzle | Near field: similar to plane jet, Far field: similar to axisymmetric jet | Aspect Ratio Dependent |
So the axisymmetric one decays faster.
Yes. In an axisymmetric jet, entrainment (entrainment of surrounding fluid) occurs from all circumferential directions, causing momentum to diffuse more rapidly.
Jet Region Structure
Let's organize the structure of a circular jet from upstream.
1. Potential Core Region ($0 < x < x_c$): The nozzle exit velocity $U_0$ is maintained at the center. $x_c \approx 4\text{--}6D$
2. Transition Region ($x_c < x < 20D$ approx.): The center velocity begins to decay
3. Self-Similar Region ($x > 20\text{--}30D$): The velocity profile becomes self-similar
The potential core length depends on the inlet turbulence intensity. Higher turbulence intensity shortens the potential core.
Self-Similar Solution
Please tell me the specific form of the self-similar solution.
In the self-similar region of an axisymmetric jet, the time-averaged velocity profile takes the following form.
The center velocity decay is derived from momentum conservation.
Here $B_u \approx 5.8\text{--}6.2$ is an experimental constant, and $x_0$ is the virtual origin. Assuming a Gaussian profile,
The value of $B_u$ varies slightly among researchers, doesn't it?
That's because it depends on the initial conditions (nozzle exit boundary layer thickness, turbulence intensity, velocity profile shape). Precise measurements by Hussein et al. (1994) reported $B_u = 5.8$, while Panchapakesan & Lumley (1993) reported $B_u = 6.06$.
Momentum Conservation
Momentum is conserved in a jet, right?
When the surroundings are a stationary fluid, the axial momentum flux remains constant.
$$ J = 2\pi \int_0^\infty \rho \bar{u}^2 r \, dr = \frac{\pi}{4} \rho U_0^2 D^2 $$
Momentum is conserved in a jet, right?
When the surroundings are a stationary fluid, the axial momentum flux remains constant.
From this relationship and the assumption of a self-similar profile, $u_c \propto x^{-1}$ and $\delta \propto x$ are derived.
Establishment of Jet Theory—From Prandtl's Mixing Length Theory to Turbulent Jets
The theoretical analysis of free jets developed based on Prandtl's (1925) mixing length theory. For a circular free jet, the similarity law holds: the centerline velocity Uc decays as Uc ∝ x⁻¹ with distance x from the jet exit, and the half-width radius increases by about 0.1 times the inlet diameter. In the 1950s-60s, Tolmien, Görtler, and others derived rigorous analytical solutions, and later the self-similarity of turbulent jets was experimentally demonstrated by the precise experiments of Wygnanski & Fiedler (1969). The discovery of this self-similarity became the tuning standard for modern RANS models, and the model constant Cμ=0.09 for k-ε was historically determined based on this experimental data.
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 in an unstable, spluttering manner, 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 due to heartbeats, or the flow fluctuation each time an engine valve opens/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 this term is set to zero. This significantly reduces computational cost, so solving first in steady-state 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 other side of the room is also because the "carrier," air, transports heat via convection. Here's the interesting part—this term contains "velocity × velocity," making it nonlinear. That is, as the flow becomes faster, 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 they naturally mix. 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 in a "thick" manner. In low Reynolds number flows (slow, viscous), diffusion is dominant. Conversely, in high Re flows, convection overwhelmingly dominates, 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 tip, right? Why? Because the piston 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 are the isobars densely packed? That's right, strong winds blow there. "Where there is a pressure difference, flow is generated"—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. When you switch to compressible analysis and suddenly get strange results, it might be due to confusing absolute/gauge pressure.
- Source Term $S_\phi$: Heated air rises—why? Because it becomes lighter (lower density) than its surroundings, so buoyancy pushes it upward. This buoyancy is added to the equation as a source term. Other examples: chemical reaction heat generated by 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 to include buoyancy results in the fluid not moving at all—a physically impossible outcome 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 (molecular 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): Treat density as constant. For Mach numbers above 0.3, compressibility effects must be considered
- Boussinesq Approximation (Natural Convection): Consider density variation only in the buoyancy term, using constant density in other terms
- Non-applicable Cases: Rarefied gases (Kn > 0.1), supersonic/hypersonic flows (shock capturing required), free surface flows (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 pressure 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 | Note confusion 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
What methods are used for jet CFD?
Jets are free shear flows, so they don't require wall resolution, making them a good match for LES.
| Method | Application Scenario | Notes |
|---|---|---|
| RANS ($k$-$\varepsilon$) | Predicting time-averaged spread rate | Note the round jet anomaly |
| RANS (SST $k$-$\omega$) | General engineering calculations | Predicts jet spread more appropriately than $k$-$\varepsilon$ |
| LES | Jet noise, detailed mixing processes | Inlet condition setting is crucial |
| DNS | Fundamental research on low Re jets | Limited to Re < $10^4$ approx. |
Round Jet Anomaly
What is the round jet anomaly?
The standard $k$-$\varepsilon$ model successfully predicts the spread rate of plane jets but overpredicts the spread rate of axisymmetric jets by about $40\%$. This is due to the $C_{\varepsilon 1}$ constant problem, stemming from the fact that the same constant cannot be used for both plane and axisymmetric jets.
Countermeasures include:
- Changing $C_{\varepsilon 1}$ from $1.44$ to $1.60$ (Pope correction)
- Using the SST $k$-$\omega$ model (improves jet spread prediction)
- Using the Realizable $k$-$\varepsilon$ model ($C_{\mu}$ becomes a variable, improving behavior for jets)
Inlet Condition Setting
How should the velocity distribution at the nozzle exit be set?
When solving jets with LES, inlet conditions greatly affect the results.
- Uniform flow profile (top-hat): Simplest but unrealistic. Lacks boundary layer at nozzle exit, altering initial shear layer development
- Pipe flow profile: $u(r) = U_c (1 - (r/R)^n)$. $n=7$ (turbulent 1/7th power law) is common
- Calculation including nozzle interior: Most accurate. Directly solves boundary layer development inside the nozzle
Injection of turbulent fluctuations is also important. Methods include:
- Synthetic Eddy Method (SEM): Jarrin et al. (2006)
- Recycling Method: Recycle data from a cross-section inside the nozzle
- Digital Filter Method: Klein et al. (2003)
So just specifying turbulence intensity isn't enough, huh.
For RANS, specifying $k$ and $\varepsilon$ (or $\omega$) at the inlet is sufficient. However, for LES, if you don't provide a spatially and temporally correlated fluctuating velocity field at the inlet, a non-physically long adaptation region occurs, shifting the potential core length.
Mesh Design
What should I be careful about with jet meshing?
The following points are important.
- Shear layer near nozzle exit: Grid size less than $1/10$ of the nozzle lip thickness is needed. To resolve the initial instability of the shear layer
- Axial domain length: At least $30D$ to see the self-similar region. For noise analysis, $50D$ or more
- Radial direction: Ensure sufficient domain ($10D$ or more) outside the jet boundary as well
- Entrainment boundary: Set pressure conditions (allowing entrainment) on side boundaries. Fixed velocity is NG
If the sides are walls, inflow can't occur, so entrainment is inhibited, right?
Exactly. If the pressure condition on the sides is incorrect, a non-physical low-pressure region occurs near the nozzle, affecting jet spread.
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