Objective Solve example and introduce Concentration Equation Discus

Objective • Solve example and introduce Concentration Equation • Discus Turbulence • Introduce Reynolds Navier Stokes Equations (RANS)

Energy equations for Newtonian fluids Conservation of energy for the fluid particle: Energy (heat) flux dissipation : source

Conservation Equations

Application Example: 2 D flow gravitation Natural convection at vertical surface - steady boundaries: -ρg 0 Twall 0 Boussinesq approximation 0 ∞ Twall>T∞ 0 0 0 0 0 0 0

Application Example: 2 D flow Driving force for natural convection

Concentration equation Conservation of mass of considered gas (chemical species): mgas=const mgas, in mgas, out dy mgas=C∙mair= C∙ρ∙dxdydz=const incompressible flow C=const dz dx Diffusion coefficient C – concentration of: H 2 O , VOC, CO 2 , and other gasses What about particles?

Turbulence Forced convection on flat plate http: //www. math. rug. nl/~veldman/cfd-gallery. html

Turbulence

Size of eddies hurricane nozzle 2 in ~200 miles Eddy ~ 1/100 in

Transition from laminar to turbulent flow

Turbulence in the vicinity of human body PT-Teknik. dk

Example The figure below shows a turbulent boundary layer due to forced convection above the flat plate. The airflow above the plate is steady-state. Consider the points A and B above the plate and line l parallel to the plate. Point A y Flow direction Point A Point B line l a) For the given time step presented on the figure above plot the velocity Vx and Vy along the line l. b) Is the stress component txy lager at point A or point B? Why? c) For point B plot the velocity Vy as function of time.

3 -D

Indoor airflow jet exhaust supply jet turbulent The question is: What we are interested in: - main flow or - turbulence?

Energy Cascade Concept in Turbulence • Kinetic energy is continually being transferred from the mean flow to the turbulent motion by large-scale eddies • The process of vortex stretching leads to a successive reduction in eddy size and to a steepening of velocity gradients between adjacent eddies. • Eventually the eddies become so small that viscous dissipation leads to the conversion of kinetic energy into heat.

Method for solving of Navier Stokes (conservation) equations • Analytical - Define boundary and initial conditions. Solve the partial deferential equations. - Solution exist for very limited number of simple cases. • Numerical - Split the considered domain into finite number of volumes (nodes). Solve the conservation equation for each volume (node). Infinitely small difference finite “small” difference

Numerical method • Simulation domain for indoor air and pollutants flow in buildings 3 D space Split or “Discretize” into smaller volumes Solve p, u, v, w, T, C

Capturing the flow properties 2” nozzle Eddy ~ 1/100 in Mesh (volume) should be smaller than eddies ! (approximately order of value)

Mesh size for direct Numerical Simulations (DNS) ~1000 ~2000 cells For 2 D wee need ~ 2 million cells Also, Turbulence is 3 -D phenomenon !

Mesh size • For 3 D simulation domain 2. 5 m Mesh size 4 m 5 m 3 D space (room) 0. 01 m → 50, 000 nodes Mesh size 0. 1 m → 50, 000 nodes 0. 001 m → 5 ∙ 1010 nodes 0. 0001 m → 5 ∙ 1013 nodes

We need to model turbulence! Reynolds Averaged Navier Stokes equations

First Methods on Analyzing Turbulent Flow - Reynolds (1895) decomposed the velocity field into a time average motion and a turbulent fluctuation vx’ Vx - Likewise f stands for any scalar: vx, vy, , vz, T, p, where: Time averaged component From this class We are going to make a difference between large and small letters

Time Averaging Operations

Averaging Navier Stokes equations Substitute into Navier Stokes equations Instantaneous velocity fluctuation around average velocity Average velocity Continuity equation: Average whole equation: Average of average = average time 0 0 Average of fluctuation = 0 0 Average