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22 August, 00:39

Consider two different flows over geometrically similar airfoil shapes, one airfoil being twice the size of the other. The flow over the smaller airfoil has freestream properties given by T[infinity] = 200 K, rho[infinity] = 1.23 kg/m3, and V[infinity] = 100 m/s. The flow over the larger airfoil is described by T[infinity] = 800 K, rho[infinity] = 1.739 kg/m3, and V[infinity] = 200 m/s. Assume that both μ and a are proportional to T1/2. Are the two flows dynamically similar?

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  1. 22 August, 03:08
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    Answer

    No;

    The two flows are not dynamically similar

    Explanation: Given

    T∞,1 = 800k

    V∞,1 = 200m/s

    p∞,1 = 1.739kg/m³

    T∞,2 = 200k

    V∞,2 = 100m/s

    p∞,2 = 1.23kg/m³

    Size1 = 2 * Size2 (L1 = 2L2) Assumptions Made

    α ∝√T

    μ∝√T Two (2) conditions must be met if the two flows are to be considered similar.

    Condition 1: Similar Parameters must be the same for both flows

    Condition 2: The bodies and boundaries must be genetically true. Condition 2 is true

    Checking for the first condition ...

    Well need to calculate Reynold's Number for both flows

    And Check if they have the same Reynold's Number Using the following formula

    Re = pVl/μ

    Re1 = p1V1l1/μ1 Re2 = p2V2l2/μ2 Re1/Re2 = p1V1l1/μ1 : p2V2l2/μ2

    Re1/Re2 = p1V1l1/μ1 * μ2/p2V2l2

    Re1/Re2 = p1V1l1μ2/p2V2l2μ1

    Re1/Re2 = p1V1l1√T2 / p1V1l1√T1

    Re1/Re2 = (1.739 * 200 * 2L2 * √200) / (1.23 * 100 * L2 * √800)

    Re1/Re2 = 9837.2/3479

    Re1/Re2 = 2,828/1

    Re1:Re2 = 2.828:1

    Re1 ≠ Re2,

    So condition 1 is not satisfied Since one of tbe conditions is not true, the two flows are not dynamically similar
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